Theory AOT_PLM

1(*<*)
2theory AOT_PLM
3  imports AOT_Axioms
4begin
5(*>*)
6
7section‹The Deductive System PLM›
8
9(* constrain sledgehammer to the abstraction layer *)
10unbundle AOT_no_atp
11
12AOT_theorem "modus-ponens": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
13  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
14lemmas MP = "modus-ponens"
15
16AOT_theorem "non-con-thm-thm": assumes ‹⊢□ φ› shows ‹⊢ φ›
17  using assms by simp
18
19AOT_theorem "vdash-properties:1[1]": assumes ‹φ ∈ Λ› shows ‹⊢ φ›
20  using assms unfolding AOT_model_act_axiom_def by blast (* NOTE: semantics needed *)
21
22text‹Convenience attribute for instantiating modally-fragile axioms.›
23attribute_setup act_axiom_inst =
24  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))›
25  "Instantiate modally fragile axiom as modally fragile theorem."
26
27AOT_theorem "vdash-properties:1[2]": assumes ‹φ ∈ Λ□› shows ‹⊢□ φ›
28  using assms unfolding AOT_model_axiom_def by blast (* NOTE: semantics needed *)
29
30text‹Convenience attribute for instantiating modally-strict axioms.›
31attribute_setup axiom_inst =
32  ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[2]"})))›
33  "Instantiate axiom as theorem."
34
35text‹Convenience methods and theorem sets for applying "cqt:2".›
36method cqt_2_lambda_inst_prover = (fast intro: AOT_instance_of_cqt_2_intro)
37method "cqt:2[lambda]" = (rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
38lemmas "cqt:2" = "cqt:2[const_var]"[axiom_inst] "cqt:2[lambda]"[axiom_inst] AOT_instance_of_cqt_2_intro
39method "cqt:2" = (safe intro!: "cqt:2")
40
41AOT_theorem "vdash-properties:3": assumes ‹⊢□ φ› shows ‹Γ ⊢ φ›
42  using assms by blast
43
44AOT_theorem "vdash-properties:5": assumes ‹Γ1 ⊢ φ› and ‹Γ2 ⊢ φ → ψ› shows ‹Γ1, Γ2 ⊢ ψ›
45  using MP assms by blast
46
47AOT_theorem "vdash-properties:6": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
48  using MP assms by blast
49
50AOT_theorem "vdash-properties:8": assumes ‹Γ ⊢ φ› and ‹φ ⊢ ψ› shows ‹Γ ⊢ ψ›
51  using assms by argo
52
53AOT_theorem "vdash-properties:9": assumes ‹φ› shows ‹ψ → φ›
54  using MP "pl:1"[axiom_inst] assms by blast
55
56AOT_theorem "vdash-properties:10": assumes ‹φ → ψ› and ‹φ› shows ‹ψ›
57  using MP assms by blast
58lemmas "→E" = "vdash-properties:10"
59
60AOT_theorem "rule-gen": assumes ‹for arbitrary α: φ{α}› shows ‹∀α φ{α}›
61  using assms by (metis AOT_var_of_term_inverse AOT_sem_denotes AOT_sem_forall) (* NOTE: semantics needed *)
62lemmas GEN = "rule-gen"
63
64AOT_theorem "RN[prem]": assumes ‹Γ ⊢□ φ› shows ‹□Γ ⊢□ □φ›
65  by (meson AOT_sem_box assms image_iff) (* NOTE: semantics needed *)
66AOT_theorem RN: assumes ‹⊢□ φ› shows ‹□φ›
67  using "RN[prem]" assms by blast
68
69AOT_axiom "df-rules-formulas[1]": assumes ‹φ ≡df ψ› shows ‹φ → ψ›
70  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
71AOT_axiom "df-rules-formulas[2]": assumes ‹φ ≡df ψ› shows ‹ψ → φ›
72  using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp) (* NOTE: semantics needed *)
73(* NOTE: for convenience also state the above as regular theorems *)
74AOT_theorem "df-rules-formulas[3]": assumes ‹φ ≡df ψ› shows ‹φ → ψ›
75  using "df-rules-formulas[1]"[axiom_inst, OF assms].
76AOT_theorem "df-rules-formulas[4]": assumes ‹φ ≡df ψ› shows ‹ψ → φ›
77  using "df-rules-formulas[2]"[axiom_inst, OF assms].
78
79
80AOT_axiom "df-rules-terms[1]":
81  assumes ‹τ{α1...αn} =df σ{α1...αn}›
82  shows ‹(σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}↓ → ¬τ{τ1...τn}↓)›
83  using assms by (simp add: AOT_model_axiomI AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
84AOT_axiom "df-rules-terms[2]":
85  assumes ‹τ =df σ›
86  shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
87  by (metis "df-rules-terms[1]" case_unit_Unity assms)
88(* NOTE: for convenience also state the above as regular theorems *)
89AOT_theorem "df-rules-terms[3]":
90  assumes ‹τ{α1...αn} =df σ{α1...αn}›
91  shows ‹(σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}) & (¬σ{τ1...τn}↓ → ¬τ{τ1...τn}↓)›
92  using "df-rules-terms[1]"[axiom_inst, OF assms].
93AOT_theorem "df-rules-terms[4]":
94  assumes ‹τ =df σ›
95  shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
96  using "df-rules-terms[2]"[axiom_inst, OF assms].
97
98
99AOT_theorem "if-p-then-p": ‹φ → φ›
100  by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)
101
102AOT_theorem "deduction-theorem": assumes ‹φ ⊢ ψ› shows ‹φ → ψ›
103  using assms by (simp add: AOT_sem_imp) (* NOTE: semantics needed *)
104lemmas CP = "deduction-theorem"
105lemmas "→I" = "deduction-theorem"
106
107AOT_theorem "ded-thm-cor:1": assumes ‹Γ1 ⊢ φ → ψ› and ‹Γ2 ⊢ ψ → χ› shows ‹Γ1, Γ2 ⊢ φ → χ›
108  using "→E" "→I" assms by blast
109AOT_theorem "ded-thm-cor:2": assumes ‹Γ1 ⊢ φ → (ψ → χ)› and ‹Γ2 ⊢ ψ› shows ‹Γ1, Γ2 ⊢ φ → χ›
110  using "→E" "→I" assms by blast
111
112AOT_theorem "ded-thm-cor:3": assumes ‹φ → ψ› and ‹ψ → χ› shows ‹φ → χ›
113  using "→E" "→I" assms by blast
114declare "ded-thm-cor:3"[trans]
115AOT_theorem "ded-thm-cor:4": assumes ‹φ → (ψ → χ)› and ‹ψ› shows ‹φ → χ›
116  using "→E" "→I" assms by blast
117
118lemmas "Hypothetical Syllogism" = "ded-thm-cor:3"
119
120AOT_theorem "useful-tautologies:1": ‹¬¬φ → φ›
121  by (metis "pl:3"[axiom_inst] "→I" "Hypothetical Syllogism")
122AOT_theorem "useful-tautologies:2": ‹φ → ¬¬φ›
123  by (metis "pl:3"[axiom_inst] "→I" "ded-thm-cor:4")
124AOT_theorem "useful-tautologies:3": ‹¬φ → (φ → ψ)›
125  by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "→I")
126AOT_theorem "useful-tautologies:4": ‹(¬ψ → ¬φ) → (φ → ψ)›
127  by (meson "pl:3"[axiom_inst] "Hypothetical Syllogism" "→I")
128AOT_theorem "useful-tautologies:5": ‹(φ → ψ) → (¬ψ → ¬φ)›
129  by (metis "useful-tautologies:4" "Hypothetical Syllogism" "→I")
130
131AOT_theorem "useful-tautologies:6": ‹(φ → ¬ψ) → (ψ → ¬φ)›
132  by (metis "→I" MP "useful-tautologies:4")
133
134AOT_theorem "useful-tautologies:7": ‹(¬φ → ψ) → (¬ψ → φ)›
135  by (metis "→I" MP "useful-tautologies:3" "useful-tautologies:5")
136
137AOT_theorem "useful-tautologies:8": ‹φ → (¬ψ → ¬(φ → ψ))›
138  by (metis "→I" MP "useful-tautologies:5")
139
140AOT_theorem "useful-tautologies:9": ‹(φ → ψ) → ((¬φ → ψ) → ψ)›
141  by (metis "→I" MP "useful-tautologies:6")
142
143AOT_theorem "useful-tautologies:10": ‹(φ → ¬ψ) → ((φ → ψ) → ¬φ)›
144  by (metis "→I" MP "pl:3"[axiom_inst])
145
146AOT_theorem "dn-i-e:1": assumes ‹φ› shows ‹¬¬φ›
147  using MP "useful-tautologies:2" assms by blast
148lemmas "¬¬I" = "dn-i-e:1"
149AOT_theorem "dn-i-e:2": assumes ‹¬¬φ› shows ‹φ›
150  using MP "useful-tautologies:1" assms by blast
151lemmas "¬¬E" = "dn-i-e:2"
152
153AOT_theorem "modus-tollens:1": assumes ‹φ → ψ› and ‹¬ψ› shows ‹¬φ›
154  using MP "useful-tautologies:5" assms by blast
155AOT_theorem "modus-tollens:2": assumes ‹φ → ¬ψ› and ‹ψ› shows ‹¬φ›
156  using "¬¬I" "modus-tollens:1" assms by blast
157lemmas MT = "modus-tollens:1" "modus-tollens:2"
158
159AOT_theorem "contraposition:1[1]": assumes ‹φ → ψ› shows ‹¬ψ → ¬φ›
160  using "→I" MT(1) assms by blast
161AOT_theorem "contraposition:1[2]": assumes ‹¬ψ → ¬φ› shows ‹φ → ψ›
162  using "→I" "¬¬E" MT(2) assms by blast
163
164AOT_theorem "contraposition:2": assumes ‹φ → ¬ψ› shows ‹ψ → ¬φ›
165  using "→I" MT(2) assms by blast
166
167(* TODO: this is actually a mixture of the two variants given in PLM; adjust. *)
168AOT_theorem "reductio-aa:1":
169  assumes ‹¬φ ⊢ ¬ψ› and ‹¬φ ⊢ ψ› shows ‹φ›
170  using "→I" "¬¬E" MT(2) assms by blast
171AOT_theorem "reductio-aa:2":
172  assumes ‹φ ⊢ ¬ψ› and ‹φ ⊢ ψ› shows ‹¬φ›
173  using "reductio-aa:1" assms by blast
174lemmas "RAA" = "reductio-aa:1" "reductio-aa:2"
175
176AOT_theorem "exc-mid": ‹φ ∨ ¬φ›
177  using "df-rules-formulas[4]" "if-p-then-p" MP "conventions:2" by blast
178
179AOT_theorem "non-contradiction": ‹¬(φ & ¬φ)›
180  using "df-rules-formulas[3]" MT(2) "useful-tautologies:2" "conventions:1" by blast
181
182AOT_theorem "con-dis-taut:1": ‹(φ & ψ) → φ›
183  by (meson "→I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
184AOT_theorem "con-dis-taut:2": ‹(φ & ψ) → ψ›
185  by (metis "→I" "df-rules-formulas[3]" MT(2) RAA(2) "¬¬E" "conventions:1")
186lemmas "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"
187
188AOT_theorem "con-dis-taut:3": ‹φ → (φ ∨ ψ)›
189  by (meson "contraposition:1[2]" "df-rules-formulas[4]" MP "→I" "conventions:2")
190AOT_theorem "con-dis-taut:4": ‹ψ → (φ ∨ ψ)›
191  using "Hypothetical Syllogism" "df-rules-formulas[4]" "pl:1"[axiom_inst] "conventions:2" by blast
192lemmas "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"
193
194AOT_theorem "con-dis-taut:5": ‹φ → (ψ → (φ & ψ))›
195  by (metis "contraposition:2" "Hypothetical Syllogism" "→I" "df-rules-formulas[4]" "conventions:1")
196lemmas Adjunction = "con-dis-taut:5"
197
198AOT_theorem "con-dis-taut:6": ‹(φ & φ) ≡ φ›
199  by (metis Adjunction "→I" "df-rules-formulas[4]" MP "Conjunction Simplification"(1) "conventions:3")
200lemmas "Idempotence of &" = "con-dis-taut:6"
201
202AOT_theorem "con-dis-taut:7": ‹(φ ∨ φ) ≡ φ›
203proof -
204  {
205    AOT_assume ‹φ ∨ φ›
206    AOT_hence ‹¬φ → φ›
207      using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
208    AOT_hence ‹φ› using "if-p-then-p" RAA(1) MP by blast
209  }
210  moreover {
211    AOT_assume ‹φ›
212    AOT_hence ‹φ ∨ φ› using "Disjunction Addition"(1) MP by blast
213  }
214  ultimately AOT_show ‹(φ ∨ φ) ≡ φ›
215    using "conventions:3"[THEN "df-rules-formulas[4]"] MP
216    by (metis Adjunction "→I")
217qed
218lemmas "Idempotence of ∨" = "con-dis-taut:7"
219
220
221AOT_theorem "con-dis-i-e:1": assumes ‹φ› and ‹ψ› shows ‹φ & ψ›
222  using Adjunction MP assms by blast
223lemmas "&I" = "con-dis-i-e:1"
224
225AOT_theorem "con-dis-i-e:2:a": assumes ‹φ & ψ› shows ‹φ›
226  using "Conjunction Simplification"(1) MP assms by blast
227AOT_theorem "con-dis-i-e:2:b": assumes ‹φ & ψ› shows ‹ψ›
228  using "Conjunction Simplification"(2) MP assms by blast
229lemmas "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"
230
231AOT_theorem "con-dis-i-e:3:a": assumes ‹φ› shows ‹φ ∨ ψ›
232  using "Disjunction Addition"(1) MP assms by blast
233AOT_theorem "con-dis-i-e:3:b": assumes ‹ψ› shows ‹φ ∨ ψ›
234  using "Disjunction Addition"(2) MP assms by blast
235AOT_theorem "con-dis-i-e:3:c": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → Θ› shows ‹χ ∨ Θ›
236  by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2) "df-rules-formulas[3]" MT(1) RAA(1) "conventions:2" assms)
237lemmas "∨I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"
238
239AOT_theorem "con-dis-i-e:4:a": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → χ› shows ‹χ›
240  by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
241AOT_theorem "con-dis-i-e:4:b": assumes ‹φ ∨ ψ› and ‹¬φ› shows ‹ψ›
242  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
243AOT_theorem "con-dis-i-e:4:c": assumes ‹φ ∨ ψ› and ‹¬ψ› shows ‹φ›
244  using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
245lemmas "∨E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"
246
247AOT_theorem "raa-cor:1": assumes ‹¬φ ⊢ ψ & ¬ψ› shows ‹φ›
248  using "&E" "∨E"(3) "∨I"(2) RAA(2) assms by blast
249AOT_theorem "raa-cor:2": assumes ‹φ ⊢ ψ & ¬ψ› shows ‹¬φ›
250  using "raa-cor:1" assms by blast
251AOT_theorem "raa-cor:3": assumes ‹φ› and ‹¬ψ ⊢ ¬φ› shows ‹ψ›
252  using RAA assms by blast
253AOT_theorem "raa-cor:4": assumes ‹¬φ› and ‹¬ψ ⊢ φ› shows ‹ψ›
254  using RAA assms by blast
255AOT_theorem "raa-cor:5": assumes ‹φ› and ‹ψ ⊢ ¬φ› shows ‹¬ψ›
256  using RAA assms by blast
257AOT_theorem "raa-cor:6": assumes ‹¬φ› and ‹ψ ⊢ φ› shows ‹¬ψ›
258  using RAA assms by blast
259
260(* TODO: note these need manual introduction rules *)
261AOT_theorem "oth-class-taut:1:a": ‹(φ → ψ) ≡ ¬(φ & ¬ψ)›
262  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
263     (metis "&E" "&I" "raa-cor:3" "→I" MP)
264AOT_theorem "oth-class-taut:1:b": ‹¬(φ → ψ) ≡ (φ & ¬ψ)›
265  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
266     (metis "&E" "&I" "raa-cor:3" "→I" MP)
267AOT_theorem "oth-class-taut:1:c": ‹(φ → ψ) ≡ (¬φ ∨ ψ)›
268  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
269     (metis "&I" "∨I"(1, 2) "∨E"(3) "→I" MP "raa-cor:1")
270
271AOT_theorem "oth-class-taut:2:a": ‹(φ & ψ) ≡ (ψ & φ)›
272  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
273     (meson "&I" "&E" "→I")
274lemmas "Commutativity of &" = "oth-class-taut:2:a"
275AOT_theorem "oth-class-taut:2:b": ‹(φ & (ψ & χ)) ≡ ((φ & ψ) & χ)›
276  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
277     (metis "&I" "&E" "→I")
278lemmas "Associativity of &" = "oth-class-taut:2:b"
279AOT_theorem "oth-class-taut:2:c": ‹(φ ∨ ψ) ≡ (ψ ∨ φ)›
280  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
281     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
282lemmas "Commutativity of ∨" = "oth-class-taut:2:c"
283AOT_theorem "oth-class-taut:2:d": ‹(φ ∨ (ψ ∨ χ)) ≡ ((φ ∨ ψ) ∨ χ)›
284  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
285     (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
286lemmas "Associativity of ∨" = "oth-class-taut:2:d"
287AOT_theorem "oth-class-taut:2:e": ‹(φ ≡ ψ) ≡ (ψ ≡ φ)›
288  by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I";
289      metis "&I" "df-rules-formulas[4]" "conventions:3" "&E" "Hypothetical Syllogism" "→I" "df-rules-formulas[3]")
290lemmas "Commutativity of ≡" = "oth-class-taut:2:e"
291AOT_theorem "oth-class-taut:2:f": ‹(φ ≡ (ψ ≡ χ)) ≡ ((φ ≡ ψ) ≡ χ)›
292  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
293        "→I" "→E" "&E" "&I"
294  by metis
295lemmas "Associativity of ≡" = "oth-class-taut:2:f"
296
297AOT_theorem "oth-class-taut:3:a": ‹φ ≡ φ›
298  using "&I" "vdash-properties:6" "if-p-then-p" "df-rules-formulas[4]" "conventions:3" by blast
299AOT_theorem "oth-class-taut:3:b": ‹φ ≡ ¬¬φ›
300  using "&I" "useful-tautologies:1" "useful-tautologies:2" "vdash-properties:6" "df-rules-formulas[4]" "conventions:3" by blast
301AOT_theorem "oth-class-taut:3:c": ‹¬(φ ≡ ¬φ)›
302  by (metis "&E" "→E" RAA "df-rules-formulas[3]" "conventions:3")
303
304AOT_theorem "oth-class-taut:4:a": ‹(φ → ψ) → ((ψ → χ) → (φ → χ))›
305  by (metis "→E" "→I")
306AOT_theorem "oth-class-taut:4:b": ‹(φ ≡ ψ) ≡ (¬φ ≡ ¬ψ)›
307  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
308        "→I" "→E" "&E" "&I" RAA by metis
309AOT_theorem "oth-class-taut:4:c": ‹(φ ≡ ψ) → ((φ → χ) ≡ (ψ → χ))›
310  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
311        "→I" "→E" "&E" "&I" by metis
312AOT_theorem "oth-class-taut:4:d": ‹(φ ≡ ψ) → ((χ → φ) ≡ (χ → ψ))›
313  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
314        "→I" "→E" "&E" "&I" by metis
315AOT_theorem "oth-class-taut:4:e": ‹(φ ≡ ψ) → ((φ & χ) ≡ (ψ & χ))›
316  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
317        "→I" "→E" "&E" "&I" by metis
318AOT_theorem "oth-class-taut:4:f": ‹(φ ≡ ψ) → ((χ & φ) ≡ (χ & ψ))›
319  using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
320        "→I" "→E" "&E" "&I" by metis
321AOT_theorem "oth-class-taut:4:g": ‹(φ ≡ ψ) ≡ ((φ & ψ) ∨ (¬φ & ¬ψ))›
322proof(safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"] "&I" "→I"
323           dest!: "conventions:3"[THEN "df-rules-formulas[3]", THEN "→E"])
324  AOT_show ‹φ & ψ ∨ (¬φ & ¬ψ)› if ‹(φ → ψ) & (ψ → φ)›
325    using "&E" "∨I" "→E" "&I" "raa-cor:1" "→I" "∨E" that by metis
326next
327  AOT_show ‹ψ› if ‹φ & ψ ∨ (¬φ & ¬ψ)› and ‹φ›
328    using that "∨E" "&E" "raa-cor:3" by blast
329next
330  AOT_show ‹φ› if ‹φ & ψ ∨ (¬φ & ¬ψ)› and ‹ψ›
331    using that "∨E" "&E" "raa-cor:3" by blast
332qed
333AOT_theorem "oth-class-taut:4:h": ‹¬(φ ≡ ψ) ≡ ((φ & ¬ψ) ∨ (¬φ & ψ))›
334proof (safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"] "&I" "→I")
335  AOT_show ‹φ & ¬ψ ∨ (¬φ & ψ)› if ‹¬(φ ≡ ψ)›
336    by (metis that "&I" "∨I"(1, 2) "→I" MT(1) "df-rules-formulas[4]" "raa-cor:3" "conventions:3")
337next
338  AOT_show ‹¬(φ ≡ ψ)› if ‹φ & ¬ψ ∨ (¬φ & ψ)›
339    by (metis that "&E" "∨E"(2) "→E" "df-rules-formulas[3]" "raa-cor:3" "conventions:3")
340qed
341AOT_theorem "oth-class-taut:5:a": ‹(φ & ψ) ≡ ¬(¬φ ∨ ¬ψ)›
342  using "conventions:3"[THEN "df-rules-formulas[4]"]
343        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
344AOT_theorem "oth-class-taut:5:b": ‹(φ ∨ ψ) ≡ ¬(¬φ & ¬ψ)›
345  using "conventions:3"[THEN "df-rules-formulas[4]"]
346        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
347AOT_theorem "oth-class-taut:5:c": ‹¬(φ & ψ) ≡ (¬φ ∨ ¬ψ)›
348  using "conventions:3"[THEN "df-rules-formulas[4]"]
349        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
350AOT_theorem "oth-class-taut:5:d": ‹¬(φ ∨ ψ) ≡ (¬φ & ¬ψ)›
351  using "conventions:3"[THEN "df-rules-formulas[4]"]
352        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
353
354lemmas DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"
355
356AOT_theorem "oth-class-taut:6:a": ‹(φ & (ψ ∨ χ)) ≡ ((φ & ψ) ∨ (φ & χ))›
357  using "conventions:3"[THEN "df-rules-formulas[4]"]
358        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
359AOT_theorem "oth-class-taut:6:b": ‹(φ ∨ (ψ & χ)) ≡ ((φ ∨ ψ) & (φ ∨ χ))›
360  using "conventions:3"[THEN "df-rules-formulas[4]"]
361        "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
362
363AOT_theorem "oth-class-taut:7:a": ‹((φ & ψ) → χ) → (φ → (ψ → χ))›
364  by (metis "&I" "→E" "→I")
365lemmas Exportation = "oth-class-taut:7:a"
366AOT_theorem "oth-class-taut:7:b": ‹(φ → (ψ →χ)) → ((φ & ψ) → χ)›
367  by (metis "&E" "→E" "→I")
368lemmas Importation = "oth-class-taut:7:b"
369
370AOT_theorem "oth-class-taut:8:a": ‹(φ → (ψ → χ)) ≡ (ψ → (φ → χ))›
371  using "conventions:3"[THEN "df-rules-formulas[4]"] "→I" "→E" "&E" "&I" by metis
372lemmas Permutation = "oth-class-taut:8:a"
373AOT_theorem "oth-class-taut:8:b": ‹(φ → ψ) → ((φ → χ) → (φ → (ψ & χ)))›
374  by (metis "&I" "→E" "→I")
375lemmas Composition = "oth-class-taut:8:b"
376AOT_theorem "oth-class-taut:8:c": ‹(φ → χ) → ((ψ → χ) → ((φ ∨ ψ) → χ))›
377  by (metis "∨E"(2) "→E" "→I" RAA(1))
378AOT_theorem "oth-class-taut:8:d": ‹((φ → ψ) & (χ → Θ)) → ((φ & χ) → (ψ & Θ))›
379  by (metis "&E" "&I" "→E" "→I")
380lemmas "Double Composition" = "oth-class-taut:8:d"
381AOT_theorem "oth-class-taut:8:e": ‹((φ & ψ) ≡ (φ & χ)) ≡ (φ → (ψ ≡ χ))›
382  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
383            "→I" "→E" "&E" "&I")
384AOT_theorem "oth-class-taut:8:f": ‹((φ & ψ) ≡ (χ & ψ)) ≡ (ψ → (φ ≡ χ))›
385  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
386            "→I" "→E" "&E" "&I")
387AOT_theorem "oth-class-taut:8:g": ‹(ψ ≡ χ) → ((φ ∨ ψ) ≡ (φ ∨ χ))›
388  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
389            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
390AOT_theorem "oth-class-taut:8:h": ‹(ψ ≡ χ) → ((ψ ∨ φ) ≡ (χ ∨ φ))›
391  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
392            "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
393AOT_theorem "oth-class-taut:8:i": ‹(φ ≡ (ψ & χ)) → (ψ → (φ ≡ χ))›
394  by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
395            "→I" "→E" "&E" "&I")
396
397AOT_theorem "intro-elim:1": assumes ‹φ ∨ ψ› and ‹φ ≡ χ› and ‹ψ ≡ Θ› shows ‹χ ∨ Θ›
398  by (metis assms "∨I"(1, 2) "∨E"(1) "conventions:3"[THEN "df-rules-formulas[3]"] "→I" "→E" "&E"(1))
399
400AOT_theorem "intro-elim:2": assumes ‹φ → ψ› and ‹ψ → φ› shows ‹φ ≡ ψ›
401  by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
402lemmas "≡I" = "intro-elim:2"
403
404AOT_theorem "intro-elim:3:a": assumes ‹φ ≡ ψ› and ‹φ› shows ‹ψ›
405  by (metis "∨I"(1) "→I" "∨E"(1) "intro-elim:1" assms)
406AOT_theorem "intro-elim:3:b": assumes ‹φ ≡ ψ› and ‹ψ› shows ‹φ›
407  using "intro-elim:3:a" "Commutativity of ≡" assms by blast
408AOT_theorem "intro-elim:3:c": assumes ‹φ ≡ ψ› and ‹¬φ› shows ‹¬ψ›
409  using "intro-elim:3:b" "raa-cor:3" assms by blast
410AOT_theorem "intro-elim:3:d": assumes ‹φ ≡ ψ› and ‹¬ψ› shows ‹¬φ›
411  using "intro-elim:3:a" "raa-cor:3" assms by blast
412AOT_theorem "intro-elim:3:e": assumes ‹φ ≡ ψ› and ‹ψ ≡ χ› shows ‹φ ≡ χ›
413  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
414declare "intro-elim:3:e"[trans]
415AOT_theorem "intro-elim:3:f": assumes ‹φ ≡ ψ› and ‹φ ≡ χ› shows ‹χ ≡ ψ›
416  by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
417lemmas "≡E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c" "intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"
418
419declare "Commutativity of ≡"[THEN "≡E"(1), sym]
420
421AOT_theorem "rule-eq-df:1": assumes ‹φ ≡df ψ› shows ‹φ ≡ ψ›
422  by (simp add: "≡I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
423lemmas "≡Df" = "rule-eq-df:1"
424AOT_theorem "rule-eq-df:2": assumes ‹φ ≡df ψ› and ‹φ› shows ‹ψ›
425  using "≡Df" "≡E"(1) assms by blast
426lemmas "≡dfE" = "rule-eq-df:2"
427AOT_theorem "rule-eq-df:3": assumes ‹φ ≡df ψ› and ‹ψ› shows ‹φ›
428  using "≡Df" "≡E"(2) assms by blast
429lemmas "≡dfI" = "rule-eq-df:3"
430
431AOT_theorem  "df-simplify:1": assumes ‹φ ≡ (ψ & χ)› and ‹ψ› shows ‹φ ≡ χ›
432  by (metis "&E"(2) "&I" "≡E"(1, 2) "≡I" "→I" assms)
433(* TODO: this is a slight variation from PLM *)
434AOT_theorem  "df-simplify:2": assumes ‹φ ≡ (ψ & χ)› and ‹χ› shows ‹φ ≡ ψ›
435  by (metis "&E"(1) "&I" "≡E"(1, 2) "≡I" "→I" assms)
436lemmas "≡S" = "df-simplify:1"  "df-simplify:2"
437
438AOT_theorem "rule-ui:1": assumes ‹∀α φ{α}› and ‹τ↓› shows ‹φ{τ}›
439  using "→E" "cqt:1"[axiom_inst] assms by blast
440AOT_theorem "rule-ui:2[const_var]": assumes ‹∀α φ{α}› shows ‹φ{β}›
441  by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
442(* TODO: precise proviso in PLM *)
443AOT_theorem "rule-ui:2[lambda]":
444  assumes ‹∀F φ{F}› and ‹INSTANCE_OF_CQT_2(ψ)›
445  shows ‹φ{[λν1...νn ψ{ν1...νn}]}›
446  by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
447AOT_theorem "rule-ui:3": assumes ‹∀α φ{α}› shows ‹φ{α}›
448  by (simp add: "rule-ui:2[const_var]" assms)
449lemmas "∀E" = "rule-ui:1" "rule-ui:2[const_var]" "rule-ui:2[lambda]" "rule-ui:3"
450
451AOT_theorem "cqt-orig:1[const_var]": ‹∀α φ{α} → φ{β}› by (simp add: "∀E"(2) "→I")
452AOT_theorem "cqt-orig:1[lambda]":
453  assumes ‹INSTANCE_OF_CQT_2(ψ)›
454  shows ‹∀F φ{F} → φ{[λν1...νn ψ{ν1...νn}]}›
455  by (simp add: "∀E"(3) "→I" assms)
456AOT_theorem "cqt-orig:2": ‹∀α (φ → ψ{α}) → (φ → ∀α ψ{α})›
457  by (metis "→I" GEN "vdash-properties:6" "∀E"(4))
458AOT_theorem "cqt-orig:3": ‹∀α φ{α} → φ{α}› using "cqt-orig:1[const_var]" .
459
460(* TODO: work out difference to GEN *)
461AOT_theorem universal: assumes ‹for arbitrary β: φ{β}› shows ‹∀α φ{α}›
462  using GEN assms .
463lemmas "∀I" = universal
464
465(* Generalized mechanism for "∀I" followed by ∀E *)
466ML‹
467fun get_instantiated_allI ctxt varname thm = let
468val trm = Thm.concl_of thm
469val trm = case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
470                      | _ => raise Term.TERM ("Expected simple theorem.", [trm])
471fun extractVars (Const (const_name‹AOT_term_of_var›, _) $ Var v) =
472    (if fst (fst v) = fst varname then [Var v] else []) (* TODO: care about the index? *)
473  | extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
474  | extractVars (Abs (_, _, t)) = extractVars t
475  | extractVars _ = []
476val vars = extractVars trm
477val vars = fold Term.add_vars vars []
478val var = hd vars
479val trmty = case (snd var) of (Type (type_name‹AOT_var›, [t])) => (t)
480              | _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
481val trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
482      Const (const_name‹AOT_term_of_var›, Type ("fun", [snd var, trmty]))
483       $ Var var, trm))
484val trm = Thm.cterm_of (Context.proof_of ctxt) trm
485val ty = hd (Term.add_tvars (Thm.prop_of @{thm "∀I"}) [])
486val typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
487val allthm = Drule.instantiate_normalize ([(ty, typ)],[]) @{thm "∀I"}
488val phi = hd (Term.add_vars (Thm.prop_of allthm) [])
489val allthm = Drule.instantiate_normalize ([],[(phi,trm)]) allthm
490in
491allthm
492end
493›
494
495attribute_setup "∀I" =
496  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
497  (fn ctxt => fn thm => fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm))›
498  "Quantify over a variable in a theorem using GEN."
499
500attribute_setup "unvarify" =
501  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
502  (fn ctxt => fn thm =>
503    let
504    val thm = fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm
505    val thm = fold (fn _ => fn thm => thm RS @{thm "∀E"(1)}) args thm
506    in
507     thm
508    end))›
509  "Generalize a statement about variables to a statement about denoting terms."
510
511(* TODO: rereplace-lem does not apply to the embedding *)
512
513AOT_theorem "cqt-basic:1": ‹∀α∀β φ{α,β} ≡ ∀β∀α φ{α,β}›
514  by (metis "≡I" "∀E"(2) "∀I" "→I")
515
516AOT_theorem "cqt-basic:2": ‹∀α(φ{α} ≡ ψ{α}) ≡ (∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α}))›
517proof (rule "≡I"; rule "→I")
518  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
519  AOT_hence ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
520  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
521    using "≡E"(1,2) "→I" by blast+
522  AOT_thus ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
523    by (auto intro: "&I" "∀I")
524next
525  AOT_assume ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
526  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
527    using "∀E"(2) "&E" by blast+
528  AOT_hence ‹φ{α} ≡ ψ{α}› for α
529    using "≡I" by blast
530  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› by (auto intro: "∀I")
531qed
532
533AOT_theorem "cqt-basic:3": ‹∀α(φ{α} ≡ ψ{α}) → (∀α φ{α} ≡ ∀α ψ{α})›
534proof(rule "→I")
535  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
536  AOT_hence 1: ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
537  {
538    AOT_assume ‹∀α φ{α}›
539    AOT_hence ‹∀α ψ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
540  }
541  moreover {
542    AOT_assume ‹∀α ψ{α}›
543    AOT_hence ‹∀α φ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
544  }
545  ultimately AOT_show ‹∀α φ{α} ≡ ∀α ψ{α}›
546    using "≡I" "→I" by auto
547qed
548
549AOT_theorem "cqt-basic:4": ‹∀α(φ{α} & ψ{α}) → (∀α φ{α} & ∀α ψ{α})›
550proof(rule "→I")
551  AOT_assume 0: ‹∀α(φ{α} & ψ{α})›
552  AOT_have ‹φ{α}› and ‹ψ{α}› for α using "∀E"(2) 0 "&E" by blast+
553  AOT_thus ‹∀α φ{α} & ∀α ψ{α}›
554    by (auto intro: "∀I" "&I")
555qed
556
557AOT_theorem "cqt-basic:5": ‹(∀α1...∀αn(φ{α1...αn})) → φ{α1...αn}›
558  using "cqt-orig:3" by blast
559
560AOT_theorem "cqt-basic:6": ‹∀α∀α φ{α} ≡ ∀α φ{α}›
561  by (meson "≡I" "→I" GEN "cqt-orig:1[const_var]")
562
563AOT_theorem "cqt-basic:7": ‹(φ → ∀α ψ{α}) ≡ ∀α(φ → ψ{α})›
564  by (metis "→I" "vdash-properties:6" "rule-ui:3" "≡I" GEN)
565
566AOT_theorem "cqt-basic:8": ‹(∀α φ{α} ∨ ∀α ψ{α}) → ∀α (φ{α} ∨ ψ{α})›
567  by (simp add: "∨I"(3) "→I" GEN "cqt-orig:1[const_var]")
568
569AOT_theorem "cqt-basic:9": ‹(∀α (φ{α} → ψ{α}) & ∀α (ψ{α} → χ{α})) → ∀α(φ{α} → χ{α})›
570proof -
571  {
572    AOT_assume ‹∀α (φ{α} → ψ{α})›
573    moreover AOT_assume ‹∀α (ψ{α} → χ{α})›
574    ultimately AOT_have ‹φ{α} → ψ{α}› and ‹ψ{α} → χ{α}› for α using "∀E" by blast+
575    AOT_hence ‹φ{α} → χ{α}› for α by (metis "→E" "→I")
576    AOT_hence ‹∀α(φ{α} → χ{α})› using "∀I" by fast
577  }
578  thus ?thesis using "&I" "→I" "&E" by meson
579qed
580
581AOT_theorem "cqt-basic:10": ‹(∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})) → ∀α (φ{α} ≡ χ{α})›
582proof(rule "→I"; rule "∀I")
583  fix β
584  AOT_assume ‹∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})›
585  AOT_hence ‹φ{β} ≡ ψ{β}› and ‹ψ{β} ≡ χ{β}› using "&E" "∀E" by blast+
586  AOT_thus ‹φ{β} ≡ χ{β}› using "≡I" "≡E" by blast
587qed
588
589AOT_theorem "cqt-basic:11": ‹∀α(φ{α} ≡ ψ{α}) ≡ ∀α (ψ{α} ≡ φ{α})›
590proof (rule "≡I"; rule "→I")
591  AOT_assume 0: ‹∀α(φ{α} ≡ ψ{α})›
592  {
593    fix α
594    AOT_have ‹φ{α} ≡ ψ{α}› using 0 "∀E" by blast
595    AOT_hence ‹ψ{α} ≡ φ{α}› using "≡I" "≡E" "→I" "→E" by metis
596  }
597  AOT_thus ‹∀α(ψ{α} ≡ φ{α})› using "∀I" by fast
598next
599  AOT_assume 0: ‹∀α(ψ{α} ≡ φ{α})›
600  {
601    fix α
602    AOT_have ‹ψ{α} ≡ φ{α}› using 0 "∀E" by blast
603    AOT_hence ‹φ{α} ≡ ψ{α}› using "≡I" "≡E" "→I" "→E" by metis
604  }
605  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› using "∀I" by fast
606qed
607
608AOT_theorem "cqt-basic:12": ‹∀α φ{α} → ∀α (ψ{α} → φ{α})›
609  by (simp add: "∀E"(2) "→I" GEN)
610
611AOT_theorem "cqt-basic:13": ‹∀α φ{α} ≡ ∀β φ{β}›
612  using "≡I" "→I" by blast
613
614AOT_theorem "cqt-basic:14": ‹(∀α1...∀αn (φ{α1...αn} → ψ{α1...αn})) → ((∀α1...∀αn φ{α1...αn}) → (∀α1...∀αn ψ{α1...αn}))›
615  using "cqt:3"[axiom_inst] by auto
616
617AOT_theorem "cqt-basic:15": ‹(∀α1...∀αn (φ → ψ{α1...αn})) → (φ → (∀α1...∀αn ψ{α1...αn}))›
618  using "cqt-orig:2" by auto
619
620(* TODO: once more the same in the embedding... need to distinguish these better *)
621AOT_theorem "universal-cor": assumes ‹for arbitrary β: φ{β}›  shows ‹∀α φ{α}›
622  using GEN assms .
623
624AOT_theorem "existential:1": assumes ‹φ{τ}› and ‹τ↓› shows ‹∃α φ{α}›
625proof(rule "raa-cor:1")
626  AOT_assume ‹¬∃α φ{α}›
627  AOT_hence ‹∀α ¬φ{α}›
628    using "≡dfI" "conventions:4" RAA "&I" by blast
629  AOT_hence ‹¬φ{τ}› using assms(2) "∀E"(1) "→E" by blast
630  AOT_thus ‹φ{τ} & ¬φ{τ}› using assms(1) "&I" by blast
631qed
632
633AOT_theorem "existential:2[const_var]": assumes ‹φ{β}› shows ‹∃α φ{α}›
634  using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast
635
636AOT_theorem "existential:2[lambda]":
637  assumes ‹φ{[λν1...νn ψ{ν1...νn}]}› and ‹INSTANCE_OF_CQT_2(ψ)›
638  shows ‹∃α φ{α}›
639  using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
640lemmas "∃I" = "existential:1" "existential:2[const_var]" "existential:2[lambda]" 
641
642AOT_theorem "instantiation":
643  assumes ‹for arbitrary β: φ{β} ⊢ ψ› and ‹∃α φ{α}›
644  shows ‹ψ›
645  by (metis (no_types, lifting) "≡dfE" GEN "raa-cor:3" "conventions:4" assms)
646lemmas "∃E" = "instantiation"
647
648AOT_theorem "cqt-further:1": ‹∀α φ{α} → ∃α φ{α}›
649  using "∀E"(4) "∃I"(2) "→I" by metis
650
651AOT_theorem "cqt-further:2": ‹¬∀α φ{α} → ∃α ¬φ{α}›
652  using "∀I" "∃I"(2) "→I" RAA by metis
653
654AOT_theorem "cqt-further:3": ‹∀α φ{α} ≡ ¬∃α ¬φ{α}›
655  using "∀E"(4) "∃E" "→I" RAA
656  by (metis "cqt-further:2" "≡I" "modus-tollens:1")
657
658AOT_theorem "cqt-further:4": ‹¬∃α φ{α} → ∀α ¬φ{α}›
659  using "∀I" "∃I"(2)"→I" RAA by metis
660
661AOT_theorem "cqt-further:5": ‹∃α (φ{α} & ψ{α}) → (∃α φ{α} & ∃α ψ{α})›
662  by (metis (no_types, lifting) "&E" "&I" "∃E" "∃I"(2) "→I")
663
664AOT_theorem "cqt-further:6": ‹∃α (φ{α} ∨ ψ{α}) → (∃α φ{α} ∨ ∃α ψ{α})›
665  by (metis (mono_tags, lifting) "∃E" "∃I"(2) "∨E"(3) "∨I"(1, 2) "→I" RAA(2))
666
667AOT_theorem "cqt-further:7": ‹∃α φ{α} ≡ ∃β φ{β}› (* TODO: vacuous in the embedding *)
668  by (simp add: "oth-class-taut:3:a")
669
670AOT_theorem "cqt-further:8": ‹(∀α φ{α} & ∀α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
671  by (metis (mono_tags, lifting) "&E" "≡I" "∀E"(2) "→I" GEN)
672
673AOT_theorem "cqt-further:9": ‹(¬∃α φ{α} & ¬∃α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
674  by (metis (mono_tags, lifting) "&E" "≡I" "∃I"(2) "→I" GEN "raa-cor:4")
675
676AOT_theorem "cqt-further:10": ‹(∃α φ{α} & ¬∃α ψ{α}) → ¬∀α (φ{α} ≡ ψ{α})›
677proof(rule "→I"; rule "raa-cor:2")
678  AOT_assume 0: ‹∃α φ{α} & ¬∃α ψ{α}›
679  then AOT_obtain α where ‹φ{α}› using "∃E" "&E"(1) by metis
680  moreover AOT_assume ‹∀α (φ{α} ≡ ψ{α})›
681  ultimately AOT_have ‹ψ{α}› using "∀E"(4) "≡E"(1) by blast
682  AOT_hence ‹∃α ψ{α}› using "∃I" by blast
683  AOT_thus ‹∃α ψ{α} & ¬∃α ψ{α}› using 0 "&E"(2) "&I" by blast
684qed
685
686AOT_theorem "cqt-further:11": ‹∃α∃β φ{α,β} ≡ ∃β∃α φ{α,β}›
687  using "≡I" "→I" "∃I"(2) "∃E" by metis
688
689AOT_theorem "log-prop-prop:1": ‹[λ φ]↓›
690  using "cqt:2[lambda0]"[axiom_inst] by auto
691
692AOT_theorem "log-prop-prop:2": ‹φ↓›
693  by (rule "≡dfI"[OF "existence:3"]) "cqt:2[lambda]"
694
695AOT_theorem "exist-nec": ‹τ↓ → □τ↓›
696proof -
697  AOT_have ‹∀β □β↓›
698    by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
699  AOT_thus ‹τ↓ → □τ↓›
700    using "cqt:1"[axiom_inst] "→E" by blast
701qed
702
703(* TODO: replace this mechanism by a "proof by types" command *)
704class AOT_Term_id = AOT_Term +
705  assumes "t=t-proper:1"[AOT]: ‹[v ⊨ τ = τ' → τ↓]›
706      and "t=t-proper:2"[AOT]: ‹[v ⊨ τ = τ' → τ'↓]›
707
708instance κ :: AOT_Term_id
709proof
710  AOT_modally_strict {
711    AOT_show ‹κ = κ' → κ↓› for κ κ'
712    proof(rule "→I")
713      AOT_assume ‹κ = κ'›
714      AOT_hence ‹O!κ ∨ A!κ›
715        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
716           (meson "→I" "∨I"(1) "&E"(1))+
717      AOT_thus ‹κ↓›
718        by (rule "∨E"(1))
719           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
720    qed
721  }
722next
723  AOT_modally_strict {
724    AOT_show ‹κ = κ' → κ'↓› for κ κ'
725    proof(rule "→I")
726      AOT_assume ‹κ = κ'›
727      AOT_hence ‹O!κ' ∨ A!κ'›
728        by (rule "∨I"(3)[OF "≡dfE"[OF "identity:1"]])
729           (meson "→I" "∨I" "&E")+
730      AOT_thus ‹κ'↓›
731        by (rule "∨E"(1))
732           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
733    qed
734  }
735qed
736
737instance rel :: (AOT_κs) AOT_Term_id
738proof
739  AOT_modally_strict {
740    AOT_show ‹Π = Π' → Π↓› for Π Π' :: ‹<'a>› (* TODO: how to get rid of the fixes? *)
741    proof(rule "→I")
742      AOT_assume ‹Π = Π'›
743      AOT_thus ‹Π↓› using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
744    qed
745  }
746next
747  AOT_modally_strict {
748    AOT_show ‹Π = Π' → Π'↓› for Π Π' :: ‹<'a>› (* TODO: how to get rid of the fixes? *)
749    proof(rule "→I")
750      AOT_assume ‹Π = Π'›
751      AOT_thus ‹Π'↓› using "≡dfE"[OF "identity:3"[of Π Π']] "&E" by blast
752    qed
753  }
754qed
755
756instance 𝗈 :: AOT_Term_id
757proof
758  AOT_modally_strict {
759    fix φ ψ
760    AOT_show ‹φ = ψ → φ↓›
761    proof(rule "→I")
762      AOT_assume ‹φ = ψ›
763      AOT_thus ‹φ↓› using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
764    qed
765  }
766next
767  AOT_modally_strict {
768    fix φ ψ
769    AOT_show ‹φ = ψ → ψ↓›
770    proof(rule "→I")
771      AOT_assume ‹φ = ψ›
772      AOT_thus ‹ψ↓› using "≡dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
773    qed
774  }
775qed
776
777instance prod :: (AOT_Term_id, AOT_Term_id) AOT_Term_id
778proof
779  AOT_modally_strict {
780    fix τ τ' :: ‹'a×'b›
781    AOT_show ‹τ = τ' → τ↓›
782    proof (induct τ; induct τ'; rule "→I")
783      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
784      AOT_assume ‹«(τ1, τ2)» = «(τ1', τ2')»›
785      AOT_hence ‹(τ1 = τ1') & (τ2 = τ2')› by (metis "≡dfE" tuple_identity_1)
786      AOT_hence ‹τ1↓› and ‹τ2↓› using "t=t-proper:1" "&E" "vdash-properties:10" by blast+
787      AOT_thus ‹«(τ1, τ2)»↓› by (metis "≡dfI" "&I" tuple_denotes)
788    qed
789  }
790next
791  AOT_modally_strict {
792    fix τ τ' :: ‹'a×'b›
793    AOT_show ‹τ = τ' → τ'↓›
794    proof (induct τ; induct τ'; rule "→I")
795      fix τ1 τ1' :: 'a and τ2  τ2' :: 'b
796      AOT_assume ‹«(τ1, τ2)» = «(τ1', τ2')»›
797      AOT_hence ‹(τ1 = τ1') & (τ2 = τ2')› by (metis "≡dfE" tuple_identity_1)
798      AOT_hence ‹τ1'↓› and ‹τ2'↓› using "t=t-proper:2" "&E" "vdash-properties:10" by blast+
799      AOT_thus ‹«(τ1', τ2')»↓› by (metis "≡dfI" "&I" tuple_denotes)
800    qed
801  }
802qed
803
804(* TODO: this is the end of the "proof by types" and makes the results available on new theorems *)
805AOT_register_type_constraints
806  Term: ‹_::AOT_Term_id› ‹_::AOT_Term_id›
807AOT_register_type_constraints
808  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id}›
809AOT_register_type_constraints
810  Relation: ‹<_::{AOT_κs, AOT_Term_id}>›
811
812AOT_theorem "id-rel-nec-equiv:1": ‹Π = Π' → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
813proof(rule "→I")
814  AOT_assume assumption: ‹Π = Π'›
815  AOT_hence ‹Π↓› and ‹Π'↓›
816    using "t=t-proper:1" "t=t-proper:2" MP by blast+
817  moreover AOT_have ‹∀F∀G (F = G → ((□∀x1...∀xn ([F]x1...xn ≡ [F]x1...xn)) → □∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)))›
818    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
819  ultimately AOT_have ‹Π = Π' → ((□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)) → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn))›
820    using "∀E"(1) by blast
821  AOT_hence ‹(□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)) → □∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
822    using assumption "→E" by blast
823  moreover AOT_have ‹□∀x1...∀xn ([Π]x1...xn ≡ [Π]x1...xn)›
824    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
825  ultimately AOT_show ‹□∀x1...∀xn ([Π]x1...xn ≡ [Π']x1...xn)›
826    using "→E" by blast
827qed
828
829AOT_theorem "id-rel-nec-equiv:2": ‹φ = ψ → □(φ ≡ ψ)›
830proof(rule "→I")
831  AOT_assume assumption: ‹φ = ψ›
832  AOT_hence ‹φ↓› and ‹ψ↓›
833    using "t=t-proper:1" "t=t-proper:2" MP by blast+
834  moreover AOT_have ‹∀p∀q (p = q → ((□(p ≡ p) → □(p ≡ q))))›
835    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
836  ultimately AOT_have ‹φ = ψ → (□(φ ≡ φ) → □(φ ≡ ψ))›
837    using "∀E"(1) by blast
838  AOT_hence ‹□(φ ≡ φ) → □(φ ≡ ψ)›
839    using assumption "→E" by blast
840  moreover AOT_have ‹□(φ ≡ φ)›
841    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
842  ultimately AOT_show ‹□(φ ≡ ψ)›
843    using "→E" by blast
844qed
845
846AOT_theorem "rule=E": assumes ‹φ{τ}› and ‹τ = σ› shows ‹φ{σ}›
847proof -
848  AOT_have ‹τ↓› and ‹σ↓› using assms(2) "t=t-proper:1" "t=t-proper:2" "→E" by blast+
849  moreover AOT_have ‹∀α∀β(α = β → (φ{α} → φ{β}))›
850    apply (rule GEN)+ using "l-identity"[axiom_inst] by blast
851  ultimately AOT_have ‹τ = σ → (φ{τ} → φ{σ})›
852    using "∀E"(1) by blast
853  AOT_thus ‹φ{σ}› using assms "→E" by blast
854qed
855
856AOT_theorem "propositions-lemma:1": ‹[λ φ] = φ›
857proof -
858  AOT_have ‹φ↓› by (simp add: "log-prop-prop:2")
859  moreover AOT_have ‹∀p [λ p] = p› using "lambda-predicates:3[zero]"[axiom_inst] "∀I" by fast
860  ultimately AOT_show ‹[λ φ] = φ›
861    using "∀E" by blast
862qed
863
864AOT_theorem "propositions-lemma:2": ‹[λ φ] ≡ φ›
865proof -
866  AOT_have ‹[λ φ] ≡ [λ φ]› by (simp add: "oth-class-taut:3:a")
867  AOT_thus ‹[λ φ] ≡ φ› using "propositions-lemma:1" "rule=E" by blast
868qed
869
870(* propositions-lemma:3 through propositions-lemma:5 do not apply *)
871
872AOT_theorem "propositions-lemma:6": ‹(φ ≡ ψ) ≡ ([λ φ] ≡ [λ ψ])›
873  by (metis "≡E"(1) "≡E"(5) "Associativity of ≡" "propositions-lemma:2")
874
875(* dr-alphabetic-rules does not apply *)
876
877AOT_theorem "oa-exist:1": ‹O!↓›
878proof -
879  AOT_have ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
880  AOT_hence 1: ‹O! = [λx ◇[E!]x]› using "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1)] "→E" by blast
881  AOT_show ‹O!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
882qed
883
884AOT_theorem "oa-exist:2": ‹A!↓›
885proof -
886  AOT_have ‹[λx ¬◇[E!]x]↓› by "cqt:2[lambda]"
887  AOT_hence 1: ‹A! = [λx ¬◇[E!]x]› using "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1)] "→E" by blast
888  AOT_show ‹A!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
889qed
890
891AOT_theorem "oa-exist:3": ‹O!x ∨ A!x›
892proof(rule "raa-cor:1")
893  AOT_assume ‹¬(O!x ∨ A!x)›
894  AOT_hence A: ‹¬O!x› and B: ‹¬A!x›
895    using "Disjunction Addition"(1) "modus-tollens:1" "∨I"(2) "raa-cor:5" by blast+
896  AOT_have C: ‹O! = [λx ◇[E!]x]›
897    by (rule "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
898  AOT_have D: ‹A! = [λx ¬◇[E!]x]›
899    by (rule "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
900  AOT_have E: ‹¬[λx ◇[E!]x]x›
901    using A C "rule=E" by fast
902  AOT_have F: ‹¬[λx ¬◇[E!]x]x›
903    using B D "rule=E" by fast
904  AOT_have G: ‹[λx ◇[E!]x]x ≡ ◇[E!]x›
905    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
906  AOT_have H: ‹[λx ¬◇[E!]x]x ≡ ¬◇[E!]x›
907    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
908  AOT_show ‹¬◇[E!]x & ¬¬◇[E!]x› using G E "≡E" H F "≡E" "&I" by metis
909qed
910
911AOT_theorem "p-identity-thm2:1": ‹F = G ≡ □∀x(x[F] ≡ x[G])›
912proof -
913  AOT_have ‹F = G ≡ F↓ & G↓ & □∀x(x[F] ≡ x[G])›
914    using "identity:2" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
915  moreover AOT_have ‹F↓› and ‹G↓›
916    by (auto simp: "cqt:2[const_var]"[axiom_inst])
917  ultimately AOT_show ‹F = G ≡ □∀x(x[F] ≡ x[G])›
918    using "≡S"(1) "&I" by blast
919qed
920
921AOT_theorem "p-identity-thm2:2[2]": ‹F = G ≡ ∀y1([λx [F]xy1] = [λx [G]xy1] & [λx [F]y1x] = [λx [G]y1x])›
922proof -
923  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1([λx [F]xy1] = [λx [G]xy1] & [λx [F]y1x] = [λx [G]y1x])›
924    using "identity:3[2]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
925  moreover AOT_have ‹F↓› and ‹G↓›
926    by (auto simp: "cqt:2[const_var]"[axiom_inst])
927  ultimately show ?thesis
928    using "≡S"(1) "&I" by blast
929qed
930    
931AOT_theorem "p-identity-thm2:2[3]": ‹F = G ≡ ∀y1∀y2([λx [F]xy1y2] = [λx [G]xy1y2] & [λx [F]y1xy2] = [λx [G]y1xy2] & [λx [F]y1y2x] = [λx [G]y1y2x])›
932proof -
933  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1∀y2([λx [F]xy1y2] = [λx [G]xy1y2] & [λx [F]y1xy2] = [λx [G]y1xy2] & [λx [F]y1y2x] = [λx [G]y1y2x])›
934    using "identity:3[3]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
935  moreover AOT_have ‹F↓› and ‹G↓›
936    by (auto simp: "cqt:2[const_var]"[axiom_inst])
937  ultimately show ?thesis
938    using "≡S"(1) "&I" by blast
939qed
940
941AOT_theorem "p-identity-thm2:2[4]": ‹F = G ≡ ∀y1∀y2∀y3([λx [F]xy1y2y3] = [λx [G]xy1y2y3] & [λx [F]y1xy2y3] = [λx [G]y1xy2y3] & [λx [F]y1y2xy3] = [λx [G]y1y2xy3] & [λx [F]y1y2y3x] = [λx [G]y1y2y3x])›
942proof -
943  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y1∀y2∀y3([λx [F]xy1y2y3] = [λx [G]xy1y2y3] & [λx [F]y1xy2y3] = [λx [G]y1xy2y3] & [λx [F]y1y2xy3] = [λx [G]y1y2xy3] & [λx [F]y1y2y3x] = [λx [G]y1y2y3x])›
944    using "identity:3[4]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
945  moreover AOT_have ‹F↓› and ‹G↓›
946    by (auto simp: "cqt:2[const_var]"[axiom_inst])
947  ultimately show ?thesis
948    using "≡S"(1) "&I" by blast
949qed
950
951AOT_theorem "p-identity-thm2:2":
952  ‹F = G ≡ ∀x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
953proof -
954  AOT_have ‹F = G ≡ F↓ & G↓ & ∀x1...∀xn «AOT_sem_proj_id x1xn (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
955    using "identity:3" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
956  moreover AOT_have ‹F↓› and ‹G↓›
957    by (auto simp: "cqt:2[const_var]"[axiom_inst])
958  ultimately show ?thesis
959    using "≡S"(1) "&I" by blast
960qed
961
962AOT_theorem "p-identity-thm2:3":
963  ‹p = q ≡ [λx p] = [λx q]›
964proof -
965  AOT_have ‹p = q ≡ p↓ & q↓ & [λx p] = [λx q]›
966    using "identity:4" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
967  moreover AOT_have ‹p↓› and ‹q↓›
968    by (auto simp: "cqt:2[const_var]"[axiom_inst])
969  ultimately show ?thesis
970    using "≡S"(1) "&I" by blast
971qed
972
973class AOT_Term_id_2 = AOT_Term_id + assumes "id-eq:1": ‹[v ⊨ α = α]›
974
975instance κ :: AOT_Term_id_2
976proof
977  AOT_modally_strict {
978    fix x
979    {
980      AOT_assume ‹O!x›
981      moreover AOT_have ‹□∀F([F]x ≡ [F]x)›
982        using RN GEN "oth-class-taut:3:a" by fast
983      ultimately AOT_have ‹O!x & O!x & □∀F([F]x ≡ [F]x)› using "&I" by simp
984    }
985    moreover {
986      AOT_assume ‹A!x›
987      moreover AOT_have ‹□∀F(x[F] ≡ x[F])›
988        using RN GEN "oth-class-taut:3:a" by fast
989      ultimately AOT_have ‹A!x & A!x & □∀F(x[F] ≡ x[F])› using "&I" by simp
990    }
991    ultimately AOT_have ‹(O!x & O!x & □∀F([F]x ≡ [F]x)) ∨ (A!x & A!x & □∀F(x[F] ≡ x[F]))›
992      using "oa-exist:3" "∨I"(1) "∨I"(2) "∨E"(3) "raa-cor:1" by blast
993    AOT_thus ‹x = x›
994      using "identity:1"[THEN "df-rules-formulas[4]"] "→E" by blast
995  }
996qed
997
998instance rel :: ("{AOT_κs,AOT_Term_id_2}") AOT_Term_id_2
999proof
1000  AOT_modally_strict {
1001    fix F :: "<'a> AOT_var"
1002    AOT_have 0: ‹[λx1...xn [F]x1...xn] = F›
1003      by (simp add: "lambda-predicates:3"[axiom_inst])
1004    AOT_have ‹[λx1...xn [F]x1...xn]↓›
1005      by "cqt:2[lambda]"
1006    AOT_hence ‹[λx1...xn [F]x1...xn] = [λx1...xn [F]x1...xn]›
1007      using "lambda-predicates:1"[axiom_inst] "→E" by blast
1008    AOT_show ‹F = F› using "rule=E" 0 by force 
1009  }
1010qed
1011
1012instance 𝗈 :: AOT_Term_id_2
1013proof
1014  AOT_modally_strict {
1015    fix p
1016    AOT_have 0: ‹[λ p] = p›
1017      by (simp add: "lambda-predicates:3[zero]"[axiom_inst])
1018    AOT_have ‹[λ p]↓›
1019      by (rule "cqt:2[lambda0]"[axiom_inst])
1020    AOT_hence ‹[λ p] = [λ p]›
1021      using "lambda-predicates:1[zero]"[axiom_inst] "→E" by blast
1022    AOT_show ‹p = p› using "rule=E" 0 by force
1023  }
1024qed
1025
1026instance prod :: (AOT_Term_id_2, AOT_Term_id_2) AOT_Term_id_2
1027proof
1028  AOT_modally_strict {
1029    fix α :: ‹('a×'b) AOT_var›
1030    AOT_show ‹α = α›
1031    proof (induct)
1032      AOT_show ‹τ = τ› if ‹τ↓› for τ :: ‹'a×'b›
1033        using that
1034      proof (induct τ)
1035        fix τ1 :: 'a and τ2 :: 'b
1036        AOT_assume ‹«(τ1,τ2)»↓›
1037        AOT_hence ‹τ1↓› and ‹τ2↓› using "≡dfE" "&E" tuple_denotes by blast+
1038        AOT_hence ‹τ1 = τ1› and ‹τ2 = τ2› using "id-eq:1"[unvarify α] by blast+
1039        AOT_thus ‹«(τ1, τ2)» = «(τ1, τ2)»› by (metis "≡dfI" "&I" tuple_identity_1)
1040      qed
1041    qed
1042  }
1043qed
1044
1045AOT_register_type_constraints
1046  Term: ‹_::AOT_Term_id_2› ‹_::AOT_Term_id_2›
1047AOT_register_type_constraints
1048  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id_2}›
1049AOT_register_type_constraints
1050  Relation: ‹<_::{AOT_κs, AOT_Term_id_2}>›
1051
1052(* TODO: Interestingly, this doesn't depend on "id-eq:1" at all! *)
1053AOT_theorem "id-eq:2": ‹α = β → β = α›
1054(*
1055  TODO: look at this proof generated using:
1056        including AOT_no_atp sledgehammer[isar_proofs = true]
1057proof -
1058  have "(∃φ. [v ⊨ ~β = α → ~φ] ∧ [v ⊨ α = β → φ]) ∨ (∃φ. ¬ [v ⊨ φ{α} → φ{β}])"
1059    by meson
1060  then show ?thesis
1061    by (meson "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" l_"identity:1" "useful-tautologies:1")
1062qed
1063*)
1064(*  by (meson "rule=E" "deduction-theorem") *)
1065proof (rule "→I")
1066  AOT_assume ‹α = β›
1067  moreover AOT_have ‹β = β› using calculation "rule=E"[of _ "λ τ . «τ = β»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1068  moreover AOT_have ‹α = α → α = α› using "if-p-then-p" by blast
1069  ultimately AOT_show ‹β = α›
1070    using "→E" "→I" "rule=E"[of _ "λ τ . «(τ = τ) → (τ = α)»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1071qed
1072
1073AOT_theorem "id-eq:3": ‹α = β & β = γ → α = γ›
1074  using "rule=E" "→I" "&E" by blast
1075
1076AOT_theorem "id-eq:4": ‹α = β ≡ ∀γ (α = γ ≡ β = γ)›
1077proof (rule "≡I"; rule "→I")
1078  AOT_assume 0: ‹α = β›
1079  AOT_hence 1: ‹β = α› using "id-eq:2" "→E" by blast
1080  AOT_show ‹∀γ (α = γ ≡ β = γ)›
1081    by (rule GEN) (metis "≡I" "→I" 0 "1" "rule=E")
1082next
1083  AOT_assume ‹∀γ (α = γ ≡ β = γ)›
1084  AOT_hence ‹α = α ≡ β = α› using "∀E"(2) by blast
1085  AOT_hence ‹α = α → β = α› using "≡E"(1) "→I" by blast
1086  AOT_hence ‹β = α› using "id-eq:1" "→E" by blast
1087  AOT_thus ‹α = β› using "id-eq:2" "→E" by blast
1088qed
1089
1090AOT_theorem "rule=I:1": assumes ‹τ↓› shows ‹τ = τ›
1091proof -
1092  AOT_have ‹∀α (α = α)›
1093    by (rule GEN) (metis "id-eq:1")
1094  AOT_thus ‹τ = τ› using assms "∀E" by blast
1095qed
1096
1097AOT_theorem "rule=I:2[const_var]": "α = α"
1098  using "id-eq:1".
1099
1100AOT_theorem "rule=I:2[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows "[λν1...νn φ{ν1...νn}] = [λν1...νn φ{ν1...νn}]"
1101proof -
1102  AOT_have ‹∀α (α = α)›
1103    by (rule GEN) (metis "id-eq:1")
1104  moreover AOT_have ‹[λν1...νn φ{ν1...νn}]↓› using assms by (rule "cqt:2[lambda]"[axiom_inst])
1105  ultimately AOT_show ‹[λν1...νn φ{ν1...νn}] = [λν1...νn φ{ν1...νn}]› using assms "∀E" by blast
1106qed
1107
1108lemmas "=I" = "rule=I:1" "rule=I:2[const_var]" "rule=I:2[lambda]"
1109
1110AOT_theorem "rule-id-df:1":
1111  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓›
1112  shows ‹τ{τ1...τn} = σ{τ1...τn}›
1113proof -
1114  AOT_have ‹σ{τ1...τn}↓ → τ{τ1...τn} = σ{τ1...τn}›
1115    using "df-rules-terms[3]" assms(1) "&E" by blast
1116  AOT_thus ‹τ{τ1...τn} = σ{τ1...τn}›
1117    using assms(2) "→E" by blast
1118qed
1119
1120AOT_theorem "rule-id-df:1[zero]":
1121  assumes ‹τ =df σ› and ‹σ↓›
1122  shows ‹τ = σ›
1123proof -
1124  AOT_have ‹σ↓ → τ = σ›
1125    using "df-rules-terms[4]" assms(1) "&E" by blast
1126  AOT_thus ‹τ = σ›
1127    using assms(2) "→E" by blast
1128qed
1129
1130AOT_theorem "rule-id-df:2:a":
1131  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓› and ‹φ{τ{τ1...τn}}›
1132  shows ‹φ{σ{τ1...τn}}›
1133proof -
1134  AOT_have ‹τ{τ1...τn} = σ{τ1...τn}› using "rule-id-df:1" assms(1,2) by blast
1135  AOT_thus ‹φ{σ{τ1...τn}}› using assms(3) "rule=E" by blast
1136qed
1137
1138(* TODO: get rid of this, ideally *)
1139AOT_theorem "rule-id-df:2:a[2]":
1140  assumes ‹τ{«(α1,α2)»} =df σ{«(α1,α2)»}› and ‹σ{«(τ1,τ2)»}↓› and ‹φ{τ{«(τ1,τ2)»}}›
1141  shows ‹φ{σ{«(τ1,τ2)»}}›
1142proof -
1143  AOT_have ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1144  proof -
1145    AOT_have ‹σ{«(τ1,τ2)»}↓ → τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1146      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1147    AOT_thus ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1148      using assms(2) "→E" by blast
1149  qed
1150  AOT_thus ‹φ{σ{«(τ1,τ2)»}}› using assms(3) "rule=E" by blast
1151qed
1152
1153AOT_theorem "rule-id-df:2:a[zero]":
1154  assumes ‹τ =df σ› and ‹σ↓› and ‹φ{τ}›
1155  shows ‹φ{σ}›
1156proof -
1157  AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
1158  AOT_thus ‹φ{σ}› using assms(3) "rule=E" by blast
1159qed
1160
1161lemmas "=dfE" = "rule-id-df:2:a" "rule-id-df:2:a[zero]"
1162
1163AOT_theorem "rule-id-df:2:b":
1164  assumes ‹τ{α1...αn} =df σ{α1...αn}› and ‹σ{τ1...τn}↓› and ‹φ{σ{τ1...τn}}›
1165  shows ‹φ{τ{τ1...τn}}›
1166proof -
1167  AOT_have ‹τ{τ1...τn} = σ{τ1...τn}› using "rule-id-df:1" assms(1,2) by blast
1168  AOT_hence ‹σ{τ1...τn} = τ{τ1...τn}›
1169    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1170  AOT_thus ‹φ{τ{τ1...τn}}› using assms(3) "rule=E" by blast
1171qed
1172
1173(* TODO: get rid of this, ideally *)
1174AOT_theorem "rule-id-df:2:b[2]":
1175  assumes ‹τ{«(α1,α2)»} =df σ{«(α1,α2)»}› and ‹σ{«(τ1,τ2)»}↓› and ‹φ{σ{«(τ1,τ2)»}}›
1176  shows ‹φ{τ{«(τ1,τ2)»}}›
1177proof -
1178  AOT_have ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1179  proof -
1180    AOT_have ‹σ{«(τ1,τ2)»}↓ → τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1181      using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def) (* NOTE: semantics needed *)
1182    AOT_thus ‹τ{«(τ1,τ2)»} = σ{«(τ1,τ2)»}›
1183      using assms(2) "→E" by blast
1184  qed
1185  AOT_hence ‹σ{«(τ1,τ2)»} = τ{«(τ1,τ2)»}›
1186    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1187  AOT_thus ‹φ{τ{«(τ1,τ2)»}}› using assms(3) "rule=E" by blast
1188qed
1189
1190AOT_theorem "rule-id-df:2:b[zero]":
1191  assumes ‹τ =df σ› and ‹σ↓› and ‹φ{σ}›
1192  shows ‹φ{τ}›
1193proof -
1194  AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
1195  AOT_hence ‹σ = τ›
1196    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1197  AOT_thus ‹φ{τ}› using assms(3) "rule=E" by blast
1198qed
1199
1200lemmas "=dfI" = "rule-id-df:2:b" "rule-id-df:2:b[zero]"
1201
1202AOT_theorem "free-thms:1": ‹τ↓ ≡ ∃β (β = τ)›
1203  by (metis "∃E" "rule=I:1" "t=t-proper:2" "→I" "∃I"(1) "≡I" "→E")
1204
1205AOT_theorem "free-thms:2": ‹∀α φ{α} → (∃β (β = τ) → φ{τ})›
1206  by (metis "∃E" "rule=E" "cqt:2[const_var]"[axiom_inst] "→I" "∀E"(1))
1207
1208AOT_theorem "free-thms:3[const_var]": ‹∃β (β = α)›
1209  by (meson "∃I"(2) "id-eq:1")
1210
1211AOT_theorem "free-thms:3[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows ‹∃β (β = [λν1...νn φ{ν1...νn}])›
1212  by (meson "=I"(3) assms "cqt:2[lambda]"[axiom_inst] "existential:1")
1213
1214AOT_theorem "free-thms:4[rel]": ‹([Π]κ1...κn ∨ κ1...κn[Π]) → ∃β (β = Π)›
1215  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1216
1217(* TODO: this is a rather weird way to formulate this and we don't have tuple-existential-elimination
1218         or tuple-equality-elimination in the theory... Splitting them is also a bit unfortunate, though.*)
1219AOT_theorem "free-thms:4[vars]": ‹([Π]κ1...κn ∨ κ1...κn[Π]) → ∃β1...∃βn (β1...βn = κ1...κn)›
1220  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1221
1222AOT_theorem "free-thms:4[1,rel]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = Π)›
1223  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1224AOT_theorem "free-thms:4[1,1]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = κ)›
1225  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1226
1227AOT_theorem "free-thms:4[2,rel]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = Π)›
1228  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1229AOT_theorem "free-thms:4[2,1]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = κ1)›
1230  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1231AOT_theorem "free-thms:4[2,2]": ‹([Π]κ1κ2 ∨ κ1κ2[Π]) → ∃β (β = κ2)›
1232  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1233AOT_theorem "free-thms:4[3,rel]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = Π)›
1234  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1235AOT_theorem "free-thms:4[3,1]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ1)›
1236  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1237AOT_theorem "free-thms:4[3,2]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ2)›
1238  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1239AOT_theorem "free-thms:4[3,3]": ‹([Π]κ1κ2κ3 ∨ κ1κ2κ3[Π]) → ∃β (β = κ3)›
1240  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1241AOT_theorem "free-thms:4[4,rel]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = Π)›
1242  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1243AOT_theorem "free-thms:4[4,1]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ1)›
1244  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1245AOT_theorem "free-thms:4[4,2]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ2)›
1246  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1247AOT_theorem "free-thms:4[4,3]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ3)›
1248  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1249AOT_theorem "free-thms:4[4,4]": ‹([Π]κ1κ2κ3κ4 ∨ κ1κ2κ3κ4[Π]) → ∃β (β = κ4)›
1250  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1251
1252AOT_theorem "ex:1:a": ‹∀α α↓›
1253  by (rule GEN) (fact "cqt:2[const_var]"[axiom_inst])
1254AOT_theorem "ex:1:b": ‹∀α∃β(β = α)›
1255  by (rule GEN) (fact "free-thms:3[const_var]")
1256
1257AOT_theorem "ex:2:a": ‹□α↓›
1258  by (rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1259AOT_theorem "ex:2:b": ‹□∃β(β = α)›
1260  by (rule RN) (fact "free-thms:3[const_var]")
1261
1262AOT_theorem "ex:3:a": ‹□∀α α↓›
1263  by (rule RN) (fact "ex:1:a")
1264AOT_theorem "ex:3:b": ‹□∀α∃β(β = α)›
1265  by (rule RN) (fact "ex:1:b")
1266
1267AOT_theorem "ex:4:a": ‹∀α □α↓›
1268  by (rule GEN; rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1269AOT_theorem "ex:4:b": ‹∀α□∃β(β = α)›
1270  by (rule GEN; rule RN) (fact "free-thms:3[const_var]")
1271
1272AOT_theorem "ex:5:a": ‹□∀α □α↓›
1273  by (rule RN) (simp add: "ex:4:a")
1274AOT_theorem "ex:5:b": ‹□∀α□∃β(β = α)›
1275  by (rule RN) (simp add: "ex:4:b")
1276
1277AOT_theorem "all-self=:1": ‹□∀α(α = α)›
1278  by (rule RN; rule GEN) (fact "id-eq:1")
1279AOT_theorem "all-self=:2": ‹∀α□(α = α)›
1280  by (rule GEN; rule RN) (fact "id-eq:1")
1281
1282AOT_theorem "id-nec:1": ‹α = β → □(α = β)›
1283proof(rule "→I")
1284  AOT_assume ‹α = β›
1285  moreover AOT_have ‹□(α = α)›
1286    by (rule RN) (fact "id-eq:1")
1287  ultimately AOT_show ‹□(α = β)› using "rule=E" by fast
1288qed
1289
1290AOT_theorem "id-nec:2": ‹τ = σ → □(τ = σ)›
1291proof(rule "→I")
1292  AOT_assume asm: ‹τ = σ›
1293  moreover AOT_have ‹τ↓›
1294    using calculation "t=t-proper:1" "→E" by blast
1295  moreover AOT_have ‹□(τ = τ)›
1296    using calculation "all-self=:2" "∀E"(1) by blast
1297  ultimately AOT_show ‹□(τ = σ)› using "rule=E" by fast
1298qed
1299
1300AOT_theorem "term-out:1": ‹φ{α} ≡ ∃β (β = α & φ{β})›
1301proof (rule "≡I"; rule "→I")
1302  AOT_assume asm: ‹φ{α}›
1303  AOT_show ‹∃β (β = α & φ{β})›
1304    by (rule "∃I"(2)[where β=α]; rule "&I")
1305       (auto simp: "id-eq:1" asm)
1306next
1307  AOT_assume 0: ‹∃β (β = α & φ{β})›
1308  (* TODO: have another look at this instantiation. Ideally AOT_obtain would resolve directly to the existential
1309           statement as proof obligation *)
1310  AOT_obtain β where ‹β = α & φ{β}› using "instantiation"[rotated, OF 0] by blast
1311  AOT_thus ‹φ{α}› using "&E" "rule=E" by blast
1312qed
1313
1314AOT_theorem "term-out:2": ‹τ↓ → (φ{τ} ≡ ∃α(α = τ & φ{α}))›
1315proof(rule "→I")
1316  AOT_assume ‹τ↓›
1317  moreover AOT_have ‹∀α (φ{α} ≡ ∃β (β = α & φ{β}))›
1318    by (rule GEN) (fact "term-out:1")
1319  ultimately AOT_show ‹φ{τ} ≡ ∃α(α = τ & φ{α})›
1320    using "∀E" by blast
1321qed
1322
1323(* TODO: example of an apply-style proof. Keep or reformulate? *)
1324AOT_theorem "term-out:3": ‹(φ{α} & ∀β(φ{β} → β = α)) ≡ ∀β(φ{β} ≡ β = α)›
1325  apply (rule "≡I"; rule "→I")
1326   apply (frule "&E"(1)) apply (drule "&E"(2))
1327   apply (rule GEN; rule "≡I"; rule "→I")
1328  using "rule-ui:2[const_var]" "vdash-properties:5" apply blast
1329  apply (meson "rule=E" "id-eq:1")
1330  apply (rule "&I")
1331  using "id-eq:1" "≡E"(2) "rule-ui:3" apply blast
1332  apply (rule GEN; rule "→I")
1333  using "≡E"(1) "rule-ui:2[const_var]" by blast
1334
1335AOT_theorem "term-out:4": ‹(φ{β} & ∀α(φ{α} → α = β)) ≡ ∀α(φ{α} ≡ α = β)›
1336  using "term-out:3" . (* TODO: same as above - another instance of the generalized alphabetic variant rule... *)
1337
1338(* TODO: would of course be nice to define it without the syntax magic *)
1339AOT_define AOT_exists_unique :: ‹α ⇒ φ ⇒ φ›
1340  "uniqueness:1": ‹«AOT_exists_unique φ» ≡df ∃α (φ{α} & ∀β (φ{β} → β = α))›
1341syntax "_AOT_exists_unique" :: ‹α ⇒ φ ⇒ φ› ("∃!_ _" [1,40])
1342AOT_syntax_print_translations
1343  "_AOT_exists_unique τ φ" <= "CONST AOT_exists_unique (_abs τ φ)"
1344syntax
1345   "_AOT_exists_unique_ellipse" :: ‹id_position ⇒ id_position ⇒ φ ⇒ φ› (‹∃!_...∃!_ _› [1,40])
1346parse_ast_translation‹[(syntax_const‹_AOT_exists_unique_ellipse›, fn ctx => fn [a,b,c] =>
1347  Ast.mk_appl (Ast.Constant "AOT_exists_unique") [parseEllipseList "_AOT_vars" ctx [a,b],c]),
1348(syntax_const‹_AOT_exists_unique›,AOT_restricted_binder const_name‹AOT_exists_unique› const_syntax‹AOT_conj›)]›
1349print_translation‹AOT_syntax_print_translations
1350  [AOT_preserve_binder_abs_tr' const_syntax‹AOT_exists_unique› syntax_const‹_AOT_exists_unique› (syntax_const‹_AOT_exists_unique_ellipse›, true) const_name‹AOT_conj›,
1351  AOT_binder_trans @{theory} @{binding "AOT_exists_unique_binder"} syntax_const‹_AOT_exists_unique›]
1352›
1353
1354
1355context AOT_meta_syntax
1356begin
1357notation AOT_exists_unique (binder "∃!" 20)
1358end
1359context AOT_no_meta_syntax
1360begin
1361no_notation AOT_exists_unique (binder "∃!" 20)
1362end
1363
1364AOT_theorem "uniqueness:2": ‹∃!α φ{α} ≡ ∃α∀β(φ{β} ≡ β = α)›
1365proof(rule "≡I"; rule "→I")
1366    AOT_assume ‹∃!α φ{α}›
1367    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1368      using "uniqueness:1" "≡dfE" by blast
1369    then AOT_obtain α where ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1370    AOT_hence ‹∀β(φ{β} ≡ β = α)› using "term-out:3" "≡E" by blast
1371    AOT_thus ‹∃α∀β(φ{β} ≡ β = α)›
1372      using "∃I" by fast
1373next
1374    AOT_assume ‹∃α∀β(φ{β} ≡ β = α)›
1375    then AOT_obtain α where ‹∀β (φ{β} ≡ β = α)› using "instantiation"[rotated] by blast
1376    AOT_hence ‹φ{α} & ∀β (φ{β} → β = α)› using "term-out:3" "≡E" by blast
1377    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1378      using "∃I" by fast
1379    AOT_thus ‹∃!α φ{α}›
1380      using "uniqueness:1" "≡dfI" by blast
1381qed
1382
1383AOT_theorem "uni-most": ‹∃!α φ{α} → ∀β∀γ((φ{β} & φ{γ}) → β = γ)›
1384proof(rule "→I"; rule GEN; rule GEN; rule "→I")
1385  fix β γ
1386  AOT_assume ‹∃!α φ{α}›
1387  AOT_hence ‹∃α∀β(φ{β} ≡ β = α)›
1388    using "uniqueness:2" "≡E" by blast
1389  then AOT_obtain α where ‹∀β(φ{β} ≡ β = α)›
1390    using "instantiation"[rotated] by blast
1391  moreover AOT_assume ‹φ{β} & φ{γ}›
1392  ultimately AOT_have ‹β = α› and ‹γ = α›
1393    using "∀E"(2) "&E" "≡E"(1,2) by blast+
1394  AOT_thus ‹β = γ›
1395    by (metis "rule=E" "id-eq:2" "→E")
1396qed
1397
1398AOT_theorem "nec-exist-!": ‹∀α(φ{α} → □φ{α}) → (∃!α φ{α} → ∃!α □φ{α})›
1399proof (rule "→I"; rule "→I")
1400  AOT_assume a: ‹∀α(φ{α} → □φ{α})›
1401  AOT_assume ‹∃!α φ{α}›
1402  AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))› using "uniqueness:1" "≡dfE" by blast
1403  then AOT_obtain α where ξ: ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1404  AOT_have ‹□φ{α}›
1405    using ξ a "&E" "∀E" "→E" by fast
1406  moreover AOT_have ‹∀β (□φ{β} → β = α)›
1407    apply (rule GEN; rule "→I")
1408    using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
1409  ultimately AOT_have ‹(□φ{α} & ∀β (□φ{β} → β = α))›
1410    using "&I" by blast
1411  AOT_thus ‹∃!α □φ{α}›
1412    using "uniqueness:1" "≡dfI" "∃I" by fast
1413qed
1414
1415AOT_theorem "act-cond": ‹𝒜(φ → ψ) → (𝒜φ → 𝒜ψ)›
1416  using "→I" "≡E"(1) "logic-actual-nec:2"[axiom_inst] by blast
1417
1418AOT_theorem "nec-imp-act": ‹□φ → 𝒜φ›
1419  by (metis "act-cond" "contraposition:1[2]" "≡E"(4) "qml:2"[THEN act_closure, axiom_inst] "qml-act:2"[axiom_inst] RAA(1) "→E" "→I")
1420
1421AOT_theorem "act-conj-act:1": ‹𝒜(𝒜φ → φ)›
1422  using "→I" "≡E"(2) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] by blast
1423
1424AOT_theorem "act-conj-act:2": ‹𝒜(φ → 𝒜φ)›
1425  by (metis "→I" "≡E"(2, 4) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] RAA(1))
1426
1427AOT_theorem "act-conj-act:3": ‹(𝒜φ & 𝒜ψ) → 𝒜(φ & ψ)›
1428proof -
1429  AOT_have ‹□(φ → (ψ → (φ & ψ)))›
1430    by (rule RN) (fact Adjunction)
1431  AOT_hence ‹𝒜(φ → (ψ → (φ & ψ)))›
1432    using "nec-imp-act" "→E" by blast
1433  AOT_hence ‹𝒜φ → 𝒜(ψ → (φ & ψ))›
1434    using "act-cond" "→E" by blast
1435  moreover AOT_have ‹𝒜(ψ → (φ & ψ)) → (𝒜ψ → 𝒜(φ & ψ))›
1436    by (fact "act-cond")
1437  ultimately AOT_have ‹𝒜φ → (𝒜ψ → 𝒜(φ & ψ))›
1438    using "→I" "→E" by metis
1439  AOT_thus ‹(𝒜φ & 𝒜ψ) → 𝒜(φ & ψ)›
1440    by (metis Importation "→E")
1441qed
1442
1443AOT_theorem "act-conj-act:4": ‹𝒜(𝒜φ ≡ φ)›
1444proof -
1445  AOT_have ‹(𝒜(𝒜φ → φ) & 𝒜(φ → 𝒜φ)) → 𝒜((𝒜φ → φ) & (φ → 𝒜φ))›
1446    by (fact "act-conj-act:3")
1447  moreover AOT_have ‹𝒜(𝒜φ → φ) & 𝒜(φ → 𝒜φ)›
1448    using "&I" "act-conj-act:1" "act-conj-act:2" by simp
1449  ultimately AOT_have ζ: ‹𝒜((𝒜φ → φ) & (φ → 𝒜φ))›
1450    using "→E" by blast
1451  AOT_have ‹𝒜(((𝒜φ → φ) & (φ → 𝒜φ)) → (𝒜φ ≡ φ))›
1452    using "conventions:3"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst] by blast
1453  AOT_hence ‹𝒜((𝒜φ → φ) & (φ → 𝒜φ)) → 𝒜(𝒜φ ≡ φ)›
1454    using "act-cond" "→E" by blast
1455  AOT_thus ‹𝒜(𝒜φ ≡ φ)› using ζ "→E" by blast
1456qed
1457
1458(* TODO: consider introducing AOT_inductive *)
1459inductive arbitrary_actualization for φ where
1460  ‹arbitrary_actualization φ «𝒜φ»›
1461| ‹arbitrary_actualization φ «𝒜ψ»› if ‹arbitrary_actualization φ ψ›
1462declare arbitrary_actualization.cases[AOT] arbitrary_actualization.induct[AOT]
1463        arbitrary_actualization.simps[AOT] arbitrary_actualization.intros[AOT]
1464syntax arbitrary_actualization :: ‹φ' ⇒ φ' ⇒ AOT_prop› ("ARBITRARY'_ACTUALIZATION'(_,_')")
1465
1466notepad
1467begin
1468  AOT_modally_strict {
1469    fix φ
1470    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜(𝒜φ ≡ φ))›
1471      using AOT_PLM.arbitrary_actualization.intros by metis
1472    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜𝒜(𝒜φ ≡ φ))›
1473      using AOT_PLM.arbitrary_actualization.intros by metis
1474    AOT_have ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, 𝒜𝒜𝒜(𝒜φ ≡ φ))›
1475      using AOT_PLM.arbitrary_actualization.intros by metis
1476  }
1477end
1478
1479
1480AOT_theorem "closure-act:1": assumes ‹ARBITRARY_ACTUALIZATION(𝒜φ ≡ φ, ψ)› shows ‹ψ›
1481using assms proof(induct)
1482  case 1
1483  AOT_show ‹𝒜(𝒜φ ≡ φ)›
1484    by (simp add: "act-conj-act:4")
1485next
1486  case (2 ψ)
1487  AOT_thus ‹𝒜ψ›
1488    by (metis arbitrary_actualization.simps "≡E"(1) "logic-actual-nec:4"[axiom_inst])
1489qed
1490
1491AOT_theorem "closure-act:2": ‹∀α 𝒜(𝒜φ{α} ≡ φ{α})›
1492  by (simp add: "act-conj-act:4" "∀I")
1493
1494AOT_theorem "closure-act:3": ‹𝒜∀α 𝒜(𝒜φ{α} ≡ φ{α})›
1495  by (metis (no_types, lifting) "act-conj-act:4" "≡E"(1,2) "logic-actual-nec:3"[axiom_inst] "logic-actual-nec:4"[axiom_inst] "∀I")
1496
1497AOT_theorem "closure-act:4": ‹𝒜∀α1...∀αn 𝒜(𝒜φ{α1...αn} ≡ φ{α1...αn})›
1498  using "closure-act:3" .
1499
1500(* TODO: examine these proofs *)
1501AOT_theorem "RA[1]": assumes ‹⊢ φ› shows ‹⊢ 𝒜φ›
1502  (* This proof is the one rejected in remark (136) (meta-rule) *)
1503  using "¬¬E" assms "≡E"(3) "logic-actual"[act_axiom_inst] "logic-actual-nec:1"[axiom_inst] "modus-tollens:2" by blast
1504AOT_theorem "RA[2]": assumes ‹⊢□ φ› shows ‹𝒜φ›
1505  (* This is actually Γ ⊢□ φ ⟹ □Γ ⊢□ 𝒜φ*)
1506  using RN assms "nec-imp-act" "vdash-properties:5" by blast
1507AOT_theorem "RA[3]": assumes ‹Γ ⊢□ φ› shows ‹𝒜Γ ⊢□ 𝒜φ›
1508  using assms by (meson AOT_sem_act imageI)
1509  (* This is not exactly right either. *)
1510
1511AOT_act_theorem "ANeg:1": ‹¬𝒜φ ≡ ¬φ›
1512  by (simp add: "RA[1]" "contraposition:1[1]" "deduction-theorem" "≡I" "logic-actual"[act_axiom_inst])
1513
1514AOT_act_theorem "ANeg:2": ‹¬𝒜¬φ ≡ φ›
1515  using "ANeg:1" "≡I" "≡E"(5) "useful-tautologies:1" "useful-tautologies:2" by blast
1516
1517AOT_theorem "Act-Basic:1": ‹𝒜φ ∨ 𝒜¬φ›
1518  by (meson "∨I"(1,2) "≡E"(2) "logic-actual-nec:1"[axiom_inst] "raa-cor:1")
1519
1520AOT_theorem "Act-Basic:2": ‹𝒜(φ & ψ) ≡ (𝒜φ & 𝒜ψ)›
1521proof (rule "≡I"; rule "→I")
1522  AOT_assume ‹𝒜(φ & ψ)›
1523  moreover AOT_have ‹𝒜((φ & ψ) → φ)›
1524    by (simp add: "RA[2]" "Conjunction Simplification"(1))
1525  moreover AOT_have ‹𝒜((φ & ψ) → ψ)›
1526    by (simp add: "RA[2]" "Conjunction Simplification"(2))
1527  ultimately AOT_show ‹𝒜φ & 𝒜ψ›
1528    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1529next
1530  AOT_assume ‹𝒜φ & 𝒜ψ›
1531  AOT_thus ‹𝒜(φ & ψ)›
1532    using "act-conj-act:3" "vdash-properties:6" by blast
1533qed
1534
1535AOT_theorem "Act-Basic:3": ‹𝒜(φ ≡ ψ) ≡ (𝒜(φ → ψ) & 𝒜(ψ → φ))›
1536proof (rule "≡I"; rule "→I")
1537  AOT_assume ‹𝒜(φ ≡ ψ)›
1538  moreover AOT_have ‹𝒜((φ ≡ ψ) → (φ → ψ))›
1539    by (simp add: "RA[2]" "deduction-theorem" "≡E"(1))
1540  moreover AOT_have ‹𝒜((φ ≡ ψ) → (ψ → φ))›
1541    by (simp add: "RA[2]" "deduction-theorem" "≡E"(2))
1542  ultimately AOT_show ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1543    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1544next
1545  AOT_assume ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1546  AOT_hence ‹𝒜((φ → ψ) & (ψ → φ))›
1547    by (metis "act-conj-act:3" "vdash-properties:10")
1548  moreover AOT_have ‹𝒜(((φ → ψ) & (ψ → φ)) → (φ ≡ ψ))›
1549    by (simp add: "conventions:3" "RA[2]" "df-rules-formulas[2]" "vdash-properties:1[2]")
1550  ultimately AOT_show ‹𝒜(φ ≡ ψ)›
1551    using "act-cond"[THEN "→E", THEN "→E"] by metis
1552qed
1553
1554AOT_theorem "Act-Basic:4": ‹(𝒜(φ → ψ) & 𝒜(ψ → φ)) ≡ (𝒜φ ≡ 𝒜ψ)›
1555proof (rule "≡I"; rule "→I")
1556  AOT_assume 0: ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1557  AOT_show ‹𝒜φ ≡ 𝒜ψ›
1558    using 0 "&E" "act-cond"[THEN "→E", THEN "→E"] "≡I" "→I" by metis
1559next
1560  AOT_assume ‹𝒜φ ≡ 𝒜ψ›
1561  AOT_thus ‹𝒜(φ → ψ) & 𝒜(ψ → φ)›
1562    by (metis "→I" "logic-actual-nec:2"[axiom_inst] "≡E"(1,2) "&I")
1563qed
1564
1565AOT_theorem "Act-Basic:5": ‹𝒜(φ ≡ ψ) ≡ (𝒜φ ≡ 𝒜ψ)›
1566  using "Act-Basic:3" "Act-Basic:4" "≡E"(5) by blast
1567
1568AOT_theorem "Act-Basic:6": ‹𝒜φ ≡ □𝒜φ›
1569  by (simp add: "≡I" "qml:2"[axiom_inst] "qml-act:1"[axiom_inst])
1570
1571AOT_theorem "Act-Basic:7": ‹𝒜□φ → □𝒜φ›
1572  by (metis "Act-Basic:6" "→I" "→E" "≡E"(1,2) "nec-imp-act" "qml-act:2"[axiom_inst])
1573
1574AOT_theorem "Act-Basic:8": ‹□φ → □𝒜φ›
1575  using "Hypothetical Syllogism" "nec-imp-act" "qml-act:1"[axiom_inst] by blast
1576
1577AOT_theorem "Act-Basic:9": ‹𝒜(φ ∨ ψ) ≡ (𝒜φ ∨ 𝒜ψ)›
1578proof (rule "≡I"; rule "→I")
1579  AOT_assume ‹𝒜(φ ∨ ψ)›
1580  AOT_thus ‹𝒜φ ∨ 𝒜ψ›
1581  proof (rule "raa-cor:3")
1582    AOT_assume ‹¬(𝒜φ ∨ 𝒜ψ)›
1583    AOT_hence ‹¬𝒜φ & ¬𝒜ψ›
1584      by (metis "≡E"(1) "oth-class-taut:5:d")
1585    AOT_hence ‹𝒜¬φ & 𝒜¬ψ›
1586      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] "&E" "&I" by metis
1587    AOT_hence ‹𝒜(¬φ & ¬ψ)›
1588      using "≡E" "Act-Basic:2" by metis
1589    moreover AOT_have ‹𝒜((¬φ & ¬ψ) ≡ ¬(φ ∨ ψ))›
1590      using "RA[2]" "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:d" by blast
1591    moreover AOT_have ‹𝒜(¬φ & ¬ψ) ≡ 𝒜(¬(φ ∨ ψ))›
1592      using calculation(2) by (metis "Act-Basic:5" "≡E"(1))
1593    ultimately AOT_have ‹𝒜(¬(φ ∨ ψ))› using "≡E" by blast
1594    AOT_thus ‹¬𝒜(φ ∨ ψ)›
1595      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by auto
1596  qed
1597next
1598  AOT_assume ‹𝒜φ ∨ 𝒜ψ›
1599  AOT_thus ‹𝒜(φ ∨ ψ)›
1600    by (meson "RA[2]" "act-cond" "∨I"(1) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
1601qed
1602
1603AOT_theorem "Act-Basic:10": ‹𝒜∃α φ{α} ≡ ∃α 𝒜φ{α}›
1604proof -
1605  AOT_have θ: ‹¬𝒜∀α ¬φ{α} ≡ ¬∀α 𝒜¬φ{α}›
1606    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1607       (metis "logic-actual-nec:3"[axiom_inst])
1608  AOT_have ξ: ‹¬∀α 𝒜¬φ{α} ≡ ¬∀α ¬𝒜φ{α}›
1609    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1610       (rule "logic-actual-nec:1"[THEN universal_closure, axiom_inst, THEN "cqt-basic:3"[THEN "→E"]])
1611  AOT_have ‹𝒜(∃α φ{α}) ≡ 𝒜(¬∀α ¬φ{α})›
1612    using "conventions:4"[THEN "df-rules-formulas[1]", THEN act_closure, axiom_inst]
1613          "conventions:4"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst]
1614    "Act-Basic:4"[THEN "≡E"(1)] "&I" "Act-Basic:5"[THEN "≡E"(2)] by metis
1615  also AOT_have ‹… ≡ ¬𝒜∀α ¬φ{α}›
1616    by (simp add: "logic-actual-nec:1" "vdash-properties:1[2]")
1617  also AOT_have ‹… ≡ ¬∀α 𝒜 ¬φ{α}› using θ by blast
1618  also AOT_have ‹… ≡ ¬∀α ¬𝒜 φ{α}› using ξ by blast
1619  also AOT_have ‹… ≡ ∃α 𝒜 φ{α}›
1620    using "conventions:4"[THEN "≡Df"] by (metis "≡E"(6) "oth-class-taut:3:a")
1621  finally AOT_show ‹𝒜∃α φ{α} ≡ ∃α 𝒜φ{α}› .
1622qed
1623
1624
1625AOT_theorem "Act-Basic:11": ‹𝒜∀α(φ{α} ≡ ψ{α}) ≡ ∀α(𝒜φ{α} ≡ 𝒜ψ{α})›
1626proof(rule "≡I"; rule "→I")
1627  AOT_assume ‹𝒜∀α(φ{α} ≡ ψ{α})›
1628  AOT_hence ‹∀α𝒜(φ{α} ≡ ψ{α})›
1629    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] by blast
1630  AOT_hence ‹𝒜(φ{α} ≡ ψ{α})› for α using "∀E" by blast
1631  AOT_hence ‹𝒜φ{α} ≡ 𝒜ψ{α}› for α by (metis "Act-Basic:5" "≡E"(1))
1632  AOT_thus ‹∀α(𝒜φ{α} ≡ 𝒜ψ{α})› by (rule "∀I")
1633next
1634  AOT_assume ‹∀α(𝒜φ{α} ≡ 𝒜ψ{α})›
1635  AOT_hence ‹𝒜φ{α} ≡ 𝒜ψ{α}› for α using "∀E" by blast
1636  AOT_hence ‹𝒜(φ{α} ≡ ψ{α})› for α by (metis "Act-Basic:5" "≡E"(2))
1637  AOT_hence ‹∀α 𝒜(φ{α} ≡ ψ{α})› by (rule "∀I")
1638  AOT_thus ‹𝒜∀α(φ{α} ≡ ψ{α})›
1639    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
1640qed
1641
1642AOT_act_theorem "act-quant-uniq": ‹∀β(𝒜φ{β} ≡ β = α) ≡ ∀β(φ{β} ≡ β = α)›
1643proof(rule "≡I"; rule "→I")
1644  AOT_assume ‹∀β(𝒜φ{β} ≡ β = α)›
1645  AOT_hence ‹𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1646  AOT_hence ‹φ{β} ≡ β = α› for β
1647    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1648    by metis
1649  AOT_thus ‹∀β(φ{β} ≡ β = α)› by (rule "∀I")
1650next
1651  AOT_assume ‹∀β(φ{β} ≡ β = α)›
1652  AOT_hence ‹φ{β} ≡ β = α› for β using "∀E" by blast
1653  AOT_hence ‹𝒜φ{β} ≡ β = α› for β
1654    using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1655    by metis
1656  AOT_thus ‹∀β(𝒜φ{β} ≡ β = α)› by (rule "∀I")
1657qed
1658
1659AOT_act_theorem "fund-cont-desc": ‹x = ιx(φ{x}) ≡ ∀z(φ{z} ≡ z = x)›
1660  using descriptions[axiom_inst] "act-quant-uniq" "≡E"(5) by fast
1661
1662AOT_act_theorem hintikka: ‹x = ιx(φ{x}) ≡ (φ{x} & ∀z (φ{z} → z = x))›
1663  using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" "fund-cont-desc" "≡E"(5) by blast
1664
1665
1666locale russel_axiom =
1667  fixes ψ
1668  assumes ψ_denotes_asm: "[v ⊨ ψ{κ}] ⟹ [v ⊨ κ↓]"
1669begin
1670AOT_act_theorem "russell-axiom": ‹ψ{ιx φ{x}} ≡ ∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1671proof -
1672  AOT_have b: ‹∀x (x = ιx φ{x} ≡ (φ{x} & ∀z(φ{z} → z = x)))›
1673    using hintikka "∀I" by fast
1674  show ?thesis
1675  proof(rule "≡I"; rule "→I")
1676    AOT_assume c: ‹ψ{ιx φ{x}}›
1677    AOT_hence d: ‹ιx φ{x}↓› using ψ_denotes_asm by blast
1678    AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1679    then AOT_obtain a where a_def: ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1680    moreover AOT_have ‹a = ιx φ{x} ≡ (φ{a} & ∀z(φ{z} → z = a))› using b "∀E" by blast
1681    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a)› using "≡E" by blast
1682    moreover AOT_have ‹ψ{a}›
1683    proof - 
1684      AOT_have 1: ‹∀x∀y(x = y → y = x)›
1685        by (simp add: "id-eq:2" "universal-cor")
1686      AOT_have ‹a = ιx φ{x} →  ιx φ{x} = a›
1687        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1688           (auto simp: 1 d "universal-cor")
1689      AOT_thus ‹ψ{a}›
1690        using a_def c "rule=E" "→E" by blast
1691    qed
1692    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a) & ψ{a}› by (rule "&I")
1693    AOT_thus ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})› by (rule "∃I")
1694  next
1695    AOT_assume ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1696    then AOT_obtain b where g: ‹φ{b} & ∀z(φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1697    AOT_hence h: ‹b = ιx φ{x} ≡ (φ{b} & ∀z(φ{z} → z = b))› using b "∀E" by blast
1698    AOT_have ‹φ{b} & ∀z(φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1699    AOT_hence ‹b = ιx φ{x}› using h "≡E" by blast
1700    AOT_thus ‹ψ{ιx φ{x}}› using j "rule=E" by blast
1701  qed
1702qed
1703end
1704
1705(* TODO: this nicely shows off using locales with the embedding, but maybe there is still a nicer way *)
1706(* TODO: sledgehammer tends to refer to ψ_denotes_asm in these instantiation instead of referring
1707         to cqt:5:a - should be fixed *)
1708interpretation "russell-axiom[exe,1]": russel_axiom ‹λ κ . «[Π]κ»›
1709  by standard (metis "cqt:5:a[1]"[axiom_inst, THEN "→E"] "&E"(2))
1710interpretation "russell-axiom[exe,2,1,1]": russel_axiom ‹λ κ . «[Π]κκ'»›
1711  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E")
1712interpretation "russell-axiom[exe,2,1,2]": russel_axiom ‹λ κ . «[Π]κ'κ»›
1713  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1714interpretation "russell-axiom[exe,2,2]": russel_axiom ‹λ κ . «[Π]κκ»›
1715  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1716interpretation "russell-axiom[exe,3,1,1]": russel_axiom ‹λ κ . «[Π]κκ'κ''»›
1717  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1718interpretation "russell-axiom[exe,3,1,2]": russel_axiom ‹λ κ . «[Π]κ'κκ''»›
1719  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1720interpretation "russell-axiom[exe,3,1,3]": russel_axiom ‹λ κ . «[Π]κ'κ''κ»›
1721  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1722interpretation "russell-axiom[exe,3,2,1]": russel_axiom ‹λ κ . «[Π]κκκ'»›
1723  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1724interpretation "russell-axiom[exe,3,2,2]": russel_axiom ‹λ κ . «[Π]κκ'κ»›
1725  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1726interpretation "russell-axiom[exe,3,2,3]": russel_axiom ‹λ κ . «[Π]κ'κκ»›
1727  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1728interpretation "russell-axiom[exe,3,3]": russel_axiom ‹λ κ . «[Π]κκκ»›
1729  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1730
1731interpretation "russell-axiom[enc,1]": russel_axiom ‹λ κ . «κ[Π]»›
1732  by standard (metis "cqt:5:b[1]"[axiom_inst, THEN "→E"] "&E"(2))
1733interpretation "russell-axiom[enc,2,1]": russel_axiom ‹λ κ . «κκ'[Π]»›
1734  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E")
1735interpretation "russell-axiom[enc,2,2]": russel_axiom ‹λ κ . «κ'κ[Π]»›
1736  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1737interpretation "russell-axiom[enc,2,3]": russel_axiom ‹λ κ . «κκ[Π]»›
1738  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1739interpretation "russell-axiom[enc,3,1,1]": russel_axiom ‹λ κ . «κκ'κ''[Π]»›
1740  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1741interpretation "russell-axiom[enc,3,1,2]": russel_axiom ‹λ κ . «κ'κκ''[Π]»›
1742  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1743interpretation "russell-axiom[enc,3,1,3]": russel_axiom ‹λ κ . «κ'κ''κ[Π]»›
1744  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1745interpretation "russell-axiom[enc,3,2,1]": russel_axiom ‹λ κ . «κκκ'[Π]»›
1746  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1747interpretation "russell-axiom[enc,3,2,2]": russel_axiom ‹λ κ . «κκ'κ[Π]»›
1748  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1749interpretation "russell-axiom[enc,3,2,3]": russel_axiom ‹λ κ . «κ'κκ[Π]»›
1750  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1751interpretation "russell-axiom[enc,3,3]": russel_axiom ‹λ κ . «κκκ[Π]»›
1752  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1753
1754AOT_act_theorem "1-exists:1": ‹ιx φ{x}↓ ≡ ∃!x φ{x}›
1755proof(rule "≡I"; rule "→I")
1756  AOT_assume ‹ιx φ{x}↓›
1757  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1758  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1759  AOT_hence ‹φ{a} & ∀z (φ{z} → z = a)› using hintikka "≡E" by blast
1760  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))› by (rule "∃I")
1761  AOT_thus ‹∃!x φ{x}› using "uniqueness:1"[THEN "≡dfI"] by blast
1762next
1763  AOT_assume ‹∃!x φ{x}›
1764  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))›
1765    using "uniqueness:1"[THEN "≡dfE"] by blast
1766  then AOT_obtain b where ‹φ{b} & ∀z (φ{z} → z = b)› using "instantiation"[rotated] by blast
1767  AOT_hence ‹b = ιx φ{x}› using hintikka "≡E" by blast
1768  AOT_thus ‹ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1769qed
1770
1771AOT_act_theorem "1-exists:2": ‹∃y(y=ιx φ{x}) ≡ ∃!x φ{x}›
1772  using "1-exists:1" "free-thms:1" "≡E"(6) by blast
1773
1774AOT_act_theorem "y-in:1": ‹x = ιx φ{x} → φ{x}›
1775  using "&E"(1) "→I" hintikka "≡E"(1) by blast
1776
1777AOT_act_theorem "y-in:2": ‹z = ιx φ{x} → φ{z}› using "y-in:1". (* TODO: same as above *)
1778
1779AOT_act_theorem "y-in:3": ‹ιx φ{x}↓ → φ{ιx φ{x}}›
1780proof(rule "→I")
1781  AOT_assume ‹ιx φ{x}↓›
1782  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1783  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1784  moreover AOT_have ‹φ{a}› using calculation hintikka "≡E"(1) "&E" by blast
1785  ultimately AOT_show ‹φ{ιx φ{x}}› using "rule=E" by blast
1786qed
1787
1788AOT_act_theorem "y-in:4": ‹∃y (y = ιx φ{x}) → φ{ιx φ{x}}›
1789  using "y-in:3"[THEN "→E"] "free-thms:1"[THEN "≡E"(2)] "→I" by blast
1790
1791
1792AOT_theorem "act-quant-nec": ‹∀β (𝒜φ{β} ≡ β = α) ≡ ∀β(𝒜𝒜φ{β} ≡ β = α)›
1793proof(rule "≡I"; rule "→I")
1794  AOT_assume ‹∀β (𝒜φ{β} ≡ β = α)›
1795  AOT_hence ‹𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1796  AOT_hence ‹𝒜𝒜φ{β} ≡ β = α› for β 
1797    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1798  AOT_thus ‹∀β(𝒜𝒜φ{β} ≡ β = α)›
1799    by (rule "∀I")
1800next
1801  AOT_assume ‹∀β(𝒜𝒜φ{β} ≡ β = α)›
1802  AOT_hence ‹𝒜𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1803  AOT_hence ‹𝒜φ{β} ≡ β = α› for β
1804    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1805  AOT_thus ‹∀β (𝒜φ{β} ≡ β = α)›
1806    by (rule "∀I")
1807qed
1808
1809AOT_theorem "equi-desc-descA:1": ‹x = ιx φ{x} ≡ x = ιx(𝒜φ{x})›
1810proof -
1811  AOT_have ‹x = ιx φ{x} ≡ ∀z (𝒜φ{z} ≡ z = x)›  using descriptions[axiom_inst] by blast
1812  also AOT_have ‹... ≡ ∀z (𝒜𝒜φ{z} ≡ z = x)›
1813  proof(rule "≡I"; rule "→I"; rule "∀I")
1814    AOT_assume ‹∀z (𝒜φ{z} ≡ z = x)›
1815    AOT_hence ‹𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1816    AOT_thus ‹𝒜𝒜φ{a} ≡ a = x› for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1817  next
1818    AOT_assume ‹∀z (𝒜𝒜φ{z} ≡ z = x)›
1819    AOT_hence ‹𝒜𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1820    AOT_thus ‹𝒜φ{a} ≡ a = x› for a  by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1821  qed
1822  also AOT_have ‹... ≡ x = ιx(𝒜φ{x})›
1823    using "Commutativity of ≡"[THEN "≡E"(1)] descriptions[axiom_inst] by fast
1824  finally show ?thesis .
1825qed
1826
1827AOT_theorem "equi-desc-descA:2": ‹ιx φ{x}↓ → ιx φ{x} = ιx(𝒜φ{x})›
1828proof(rule "→I")
1829  AOT_assume ‹ιx φ{x}↓›
1830  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1831  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1832  moreover AOT_have ‹a = ιx(𝒜φ{x})› using calculation "equi-desc-descA:1"[THEN "≡E"(1)] by blast
1833  ultimately AOT_show ‹ιx φ{x} = ιx(𝒜φ{x})› using "rule=E" by fast
1834qed
1835
1836AOT_theorem "nec-hintikka-scheme": ‹x = ιx φ{x} ≡ 𝒜φ{x} & ∀z(𝒜φ{z} → z = x)›
1837proof -
1838  AOT_have ‹x = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = x)› using descriptions[axiom_inst] by blast
1839  also AOT_have ‹… ≡ (𝒜φ{x} & ∀z(𝒜φ{z} → z = x))›
1840    using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" by fast
1841  finally show ?thesis.
1842qed
1843
1844AOT_theorem "equiv-desc-eq:1": ‹𝒜∀x(φ{x} ≡ ψ{x}) → ∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1845proof(rule "→I"; rule "∀I")
1846  fix β
1847  AOT_assume ‹𝒜∀x(φ{x} ≡ ψ{x})›
1848  AOT_hence ‹𝒜(φ{x} ≡ ψ{x})› for x using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] "∀E"(2) by blast
1849  AOT_hence 0: ‹𝒜φ{x} ≡ 𝒜ψ{x}› for x by (metis "Act-Basic:5" "≡E"(1))
1850  AOT_have ‹β = ιx φ{x} ≡ 𝒜φ{β} & ∀z(𝒜φ{z} → z = β)› using "nec-hintikka-scheme" by blast
1851  also AOT_have ‹... ≡ 𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)›
1852  proof (rule "≡I"; rule "→I")
1853    AOT_assume 1: ‹𝒜φ{β} & ∀z(𝒜φ{z} → z = β)›
1854    AOT_hence ‹𝒜φ{z} → z = β› for z using "&E" "∀E" by blast
1855    AOT_hence ‹𝒜ψ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1856    AOT_hence ‹∀z(𝒜ψ{z} → z = β)› using "∀I" by fast
1857    moreover AOT_have ‹𝒜ψ{β}› using "&E" 0[THEN "≡E"(1)] 1 by blast
1858    ultimately AOT_show ‹𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)› using "&I" by blast
1859  next
1860    AOT_assume 1: ‹𝒜ψ{β} & ∀z(𝒜ψ{z} → z = β)›
1861    AOT_hence ‹𝒜ψ{z} → z = β› for z using "&E" "∀E" by blast
1862    AOT_hence ‹𝒜φ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1863    AOT_hence ‹∀z(𝒜φ{z} → z = β)› using "∀I" by fast
1864    moreover AOT_have ‹𝒜φ{β}› using "&E" 0[THEN "≡E"(2)] 1 by blast
1865    ultimately AOT_show ‹𝒜φ{β} & ∀z(𝒜φ{z} → z = β)› using "&I" by blast
1866  qed
1867  also AOT_have ‹... ≡ β = ιx ψ{x}›
1868    using "Commutativity of ≡"[THEN "≡E"(1)] "nec-hintikka-scheme" by blast
1869  finally AOT_show ‹β = ιx φ{x} ≡ β = ιx ψ{x}› .
1870qed
1871
1872AOT_theorem "equiv-desc-eq:2": ‹ιx φ{x}↓ & 𝒜∀x(φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1873proof(rule "→I")
1874  AOT_assume ‹ιx φ{x}↓ & 𝒜∀x(φ{x} ≡ ψ{x})›
1875  AOT_hence 0: ‹∃y (y = ιx φ{x})› and
1876            1: ‹∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1877    using "&E" "free-thms:1"[THEN "≡E"(1)] "equiv-desc-eq:1" "→E" by blast+
1878  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1879  moreover AOT_have ‹a = ιx ψ{x}› using calculation 1 "∀E" "≡E"(1) by fast
1880  ultimately AOT_show ‹ιx φ{x} = ιx ψ{x}›
1881    using "rule=E" by fast
1882qed
1883
1884AOT_theorem "equiv-desc-eq:3": ‹ιx φ{x}↓ & □∀x(φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1885  using "→I" "equiv-desc-eq:2"[THEN "→E", OF "&I"] "&E" "nec-imp-act"[THEN "→E"] by metis
1886
1887(* Note: this is a special case of "exist-nec" *)
1888AOT_theorem "equiv-desc-eq:4": ‹ιx φ{x}↓ → □ιx φ{x}↓›
1889proof(rule "→I")
1890  AOT_assume ‹ιx φ{x}↓›
1891  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1892  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1893  AOT_thus ‹□ιx φ{x}↓›
1894    using "ex:2:a" "rule=E" by fast
1895qed
1896
1897AOT_theorem "equiv-desc-eq:5": ‹ιx φ{x}↓ → ∃y □(y = ιx φ{x})›
1898proof(rule "→I")
1899  AOT_assume ‹ιx φ{x}↓›
1900  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1901  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1902  AOT_hence ‹□(a = ιx φ{x})› by (metis "id-nec:2" "vdash-properties:10")
1903  AOT_thus ‹∃y □(y = ιx φ{x})› by (rule "∃I")
1904qed
1905
1906AOT_act_theorem "equiv-desc-eq2:1": ‹∀x (φ{x} ≡ ψ{x}) → ∀x (x = ιx φ{x} ≡ x = ιx ψ{x})›
1907  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:1"[THEN "→E"]
1908        "RA[1]" "deduction-theorem" by blast
1909
1910AOT_act_theorem "equiv-desc-eq2:2": ‹ιx φ{x}↓ & ∀x (φ{x} ≡ ψ{x}) → ιx φ{x} = ιx ψ{x}›
1911  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:2"[THEN "→E", OF "&I"]
1912        "RA[1]" "deduction-theorem" "&E" by metis
1913
1914context russel_axiom
1915begin
1916AOT_theorem "nec-russell-axiom": ‹ψ{ιx φ{x}} ≡ ∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})›
1917proof -
1918  AOT_have b: ‹∀x (x = ιx φ{x} ≡ (𝒜φ{x} & ∀z(𝒜φ{z} → z = x)))›
1919    using "nec-hintikka-scheme" "∀I" by fast
1920  show ?thesis
1921  proof(rule "≡I"; rule "→I")
1922    AOT_assume c: ‹ψ{ιx φ{x}}›
1923    AOT_hence d: ‹ιx φ{x}↓› using ψ_denotes_asm by blast
1924    AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1925    then AOT_obtain a where a_def: ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1926    moreover AOT_have ‹a = ιx φ{x} ≡ (𝒜φ{a} & ∀z(𝒜φ{z} → z = a))› using b "∀E" by blast
1927    ultimately AOT_have ‹𝒜φ{a} & ∀z(𝒜φ{z} → z = a)› using "≡E" by blast
1928    moreover AOT_have ‹ψ{a}›
1929    proof - 
1930      AOT_have 1: ‹∀x∀y(x = y → y = x)›
1931        by (simp add: "id-eq:2" "universal-cor")
1932      AOT_have ‹a = ιx φ{x} →  ιx φ{x} = a›
1933        by (rule "∀E"(1)[where τ="«ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1934           (auto simp: d "universal-cor" 1)
1935      AOT_thus ‹ψ{a}›
1936        using a_def c "rule=E" "→E" by metis
1937    qed
1938    ultimately AOT_have ‹𝒜φ{a} & ∀z(𝒜φ{z} → z = a) & ψ{a}› by (rule "&I")
1939    AOT_thus ‹∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})› by (rule "∃I")
1940  next
1941    AOT_assume ‹∃x(𝒜φ{x} & ∀z(𝒜φ{z} → z = x) & ψ{x})›
1942    then AOT_obtain b where g: ‹𝒜φ{b} & ∀z(𝒜φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1943    AOT_hence h: ‹b = ιx φ{x} ≡ (𝒜φ{b} & ∀z(𝒜φ{z} → z = b))› using b "∀E" by blast
1944    AOT_have ‹𝒜φ{b} & ∀z(𝒜φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1945    AOT_hence ‹b = ιx φ{x}› using h "≡E" by blast
1946    AOT_thus ‹ψ{ιx φ{x}}› using j "rule=E" by blast
1947  qed
1948qed
1949end
1950
1951AOT_theorem "actual-desc:1": ‹ιx φ{x}↓ ≡ ∃!x 𝒜φ{x}›
1952proof (rule "≡I"; rule "→I")
1953  AOT_assume ‹ιx φ{x}↓›
1954  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1955  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1956  moreover AOT_have ‹a = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = a)›
1957    using descriptions[axiom_inst] by blast
1958  ultimately AOT_have ‹∀z(𝒜φ{z} ≡ z = a)›
1959    using "≡E" by blast
1960  AOT_hence ‹∃x∀z(𝒜φ{z} ≡ z = x)› by (rule "∃I")
1961  AOT_thus ‹∃!x 𝒜φ{x}›
1962    using "uniqueness:2"[THEN "≡E"(2)] by fast
1963next
1964  AOT_assume ‹∃!x 𝒜φ{x}›
1965  AOT_hence ‹∃x∀z(𝒜φ{z} ≡ z = x)›
1966    using "uniqueness:2"[THEN "≡E"(1)] by fast
1967  then AOT_obtain a where ‹∀z(𝒜φ{z} ≡ z = a)› using "instantiation"[rotated] by blast
1968  moreover AOT_have ‹a = ιx φ{x} ≡ ∀z(𝒜φ{z} ≡ z = a)›
1969    using descriptions[axiom_inst] by blast
1970  ultimately AOT_have ‹a = ιx φ{x}› using "≡E" by blast
1971  AOT_thus ‹ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1972qed
1973
1974AOT_theorem "actual-desc:2": ‹x = ιx φ{x} → 𝒜φ{x}›
1975  using "&E"(1) "contraposition:1[2]" "≡E"(1) "nec-hintikka-scheme" "reductio-aa:2" "vdash-properties:9" by blast
1976
1977AOT_theorem "actual-desc:3": ‹z = ιx φ{x} → 𝒜φ{z}›
1978  using "actual-desc:2". (* TODO: same as above *)
1979
1980AOT_theorem "actual-desc:4": ‹ιx φ{x}↓ → 𝒜φ{ιx φ{x}}›
1981proof(rule "→I")
1982  AOT_assume ‹ιx φ{x}↓›
1983  AOT_hence ‹∃y (y = ιx φ{x})› by (metis "rule=I:1" "existential:1")
1984  then AOT_obtain a where ‹a = ιx φ{x}› using "instantiation"[rotated] by blast
1985  AOT_thus ‹𝒜φ{ιx φ{x}}›
1986    using "actual-desc:2" "rule=E" "→E" by fast
1987qed
1988
1989(* TODO: take another look at proof in PLM *)
1990AOT_theorem "actual-desc:5": ‹ιx φ{x} = ιx ψ{x} → 𝒜∀x(φ{x} ≡ ψ{x})›
1991proof(rule "→I")
1992  AOT_assume 0: ‹ιx φ{x} = ιx ψ{x}›
1993  AOT_hence φ_down: ‹ιx φ{x}↓› and ψ_down: ‹ιx ψ{x}↓›
1994    using "t=t-proper:1" "t=t-proper:2" "vdash-properties:6" by blast+
1995  AOT_hence ‹∃y (y = ιx φ{x})› and ‹∃y (y = ιx ψ{x})› by (metis "rule=I:1" "existential:1")+
1996  then AOT_obtain a and b where a_eq: ‹a = ιx φ{x}› and b_eq: ‹b = ιx ψ{x}›
1997    using "instantiation"[rotated] by metis
1998
1999  AOT_have ‹∀α∀β (α = β → β = α)› by (rule "∀I"; rule "∀I"; rule "id-eq:2")
2000  AOT_hence ‹∀β (ιx φ{x} = β → β = ιx φ{x})›
2001    using "∀E" φ_down by blast
2002  AOT_hence ‹ιx φ{x} = ιx ψ{x} → ιx ψ{x} = ιx φ{x}›
2003    using "∀E" ψ_down by blast
2004  AOT_hence 1: ‹ιx ψ{x} = ιx φ{x}› using 0
2005    "→E" by blast
2006
2007  AOT_have ‹𝒜φ{x} ≡ 𝒜ψ{x}› for x
2008  proof(rule "≡I"; rule "→I")
2009    AOT_assume ‹𝒜φ{x}›
2010    moreover AOT_have ‹𝒜φ{x} → x = a› for x
2011      using "nec-hintikka-scheme"[THEN "≡E"(1), OF a_eq, THEN "&E"(2)] "∀E" by blast
2012    ultimately AOT_have ‹x = a› using "→E" by blast
2013    AOT_hence ‹x = ιx φ{x}› using a_eq "rule=E" by blast
2014    AOT_hence ‹x = ιx ψ{x}› using 0 "rule=E" by blast
2015    AOT_thus ‹𝒜ψ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2016  next
2017    AOT_assume ‹𝒜ψ{x}›
2018    moreover AOT_have ‹𝒜ψ{x} → x = b› for x
2019      using "nec-hintikka-scheme"[THEN "≡E"(1), OF b_eq, THEN "&E"(2)] "∀E" by blast
2020    ultimately AOT_have ‹x = b› using "→E" by blast
2021    AOT_hence ‹x = ιx ψ{x}› using b_eq "rule=E" by blast
2022    AOT_hence ‹x = ιx φ{x}› using 1 "rule=E" by blast
2023    AOT_thus ‹𝒜φ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2024  qed
2025  AOT_hence ‹𝒜(φ{x} ≡ ψ{x})› for x by (metis "Act-Basic:5" "≡E"(2))
2026  AOT_hence ‹∀x 𝒜(φ{x} ≡ ψ{x})› by (rule "∀I")
2027  AOT_thus ‹𝒜∀x (φ{x} ≡ ψ{x})›
2028    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
2029qed    
2030
2031AOT_theorem "!box-desc:1": ‹∃!x □φ{x} → ∀y (y = ιx φ{x} → φ{y})›
2032proof(rule "→I")
2033  AOT_assume ‹∃!x □φ{x}›
2034  AOT_hence ζ: ‹∃x (□φ{x} & ∀z (□φ{z} → z = x))›
2035    using "uniqueness:1"[THEN "≡dfE"] by blast
2036  then AOT_obtain b where θ: ‹□φ{b} & ∀z (□φ{z} → z = b)› using "instantiation"[rotated] by blast
2037  AOT_show ‹∀y (y = ιx φ{x} → φ{y})›
2038  proof(rule GEN; rule "→I")
2039    fix y
2040    AOT_assume ‹y = ιx φ{x}›
2041    AOT_hence ‹𝒜φ{y} & ∀z (𝒜φ{z} → z = y)› using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
2042    AOT_hence ‹𝒜φ{b} → b = y› using "&E" "∀E" by blast
2043    moreover AOT_have ‹𝒜φ{b}› using θ[THEN "&E"(1)]  by (metis "nec-imp-act" "→E")
2044    ultimately AOT_have ‹b = y› using "→E" by blast
2045    moreover AOT_have ‹φ{b}› using θ[THEN "&E"(1)]  by (metis "qml:2"[axiom_inst] "→E") 
2046    ultimately AOT_show ‹φ{y}› using "rule=E" by blast
2047  qed
2048qed
2049
2050AOT_theorem "!box-desc:2": ‹∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ∀y (y = ιx φ{x} → φ{y}))›
2051proof(rule "→I"; rule "→I")
2052  AOT_assume ‹∀x (φ{x} → □φ{x})›
2053  moreover AOT_assume ‹∃!x φ{x}›
2054  ultimately AOT_have ‹∃!x □φ{x}›
2055    using "nec-exist-!"[THEN "→E", THEN "→E"] by blast
2056  AOT_thus ‹∀y (y = ιx φ{x} → φ{y})›
2057    using "!box-desc:1" "→E" by blast
2058qed
2059
2060AOT_theorem "dr-alphabetic-thm": ‹ιν φ{ν}↓ → ιν φ{ν} = ιμ φ{μ}› (* TODO: vacuous *)
2061  by (simp add: "rule=I:1" "→I")
2062
2063AOT_theorem "RM:1[prem]": assumes ‹Γ ⊢□ φ → ψ› shows ‹□Γ ⊢□ □φ → □ψ›
2064proof -
2065  AOT_have ‹□Γ ⊢□ □(φ → ψ)› using "RN[prem]" assms by blast
2066  AOT_thus ‹□Γ ⊢□ □φ → □ψ› by (metis "qml:1"[axiom_inst] "→E")
2067qed
2068
2069AOT_theorem "RM:1": assumes ‹⊢□ φ → ψ› shows ‹⊢□ □φ → □ψ›
2070  using "RM:1[prem]" assms by blast
2071
2072lemmas RM = "RM:1"
2073
2074AOT_theorem "RM:2[prem]": assumes ‹Γ ⊢□ φ → ψ› shows ‹□Γ ⊢□ ◇φ → ◇ψ›
2075proof -
2076  AOT_have ‹Γ ⊢□ ¬ψ → ¬φ› using assms 
2077    by (simp add: "contraposition:1[1]")
2078  AOT_hence ‹□Γ ⊢□ □¬ψ → □¬φ› using "RM:1[prem]" by blast
2079  AOT_thus ‹□Γ ⊢□ ◇φ → ◇ψ›
2080    by (meson "≡dfE" "≡dfI" "conventions:5" "deduction-theorem" "modus-tollens:1")
2081qed
2082
2083AOT_theorem "RM:2": assumes ‹⊢□ φ → ψ› shows ‹⊢□ ◇φ → ◇ψ›
2084  using "RM:2[prem]" assms by blast
2085
2086lemmas "RM◇" = "RM:2"
2087
2088AOT_theorem "RM:3[prem]": assumes ‹Γ ⊢□ φ ≡ ψ› shows ‹□Γ ⊢□ □φ ≡ □ψ›
2089proof -
2090  AOT_have ‹Γ ⊢□ φ → ψ› and ‹Γ ⊢□ ψ → φ› using assms "≡E" "→I" by metis+
2091  AOT_hence ‹□Γ ⊢□ □φ → □ψ› and ‹□Γ ⊢□ □ψ → □φ› using "RM:1[prem]" by metis+
2092  AOT_thus ‹□Γ ⊢□ □φ ≡ □ψ›
2093    by (simp add: "≡I")
2094qed
2095
2096AOT_theorem "RM:3": assumes ‹⊢□ φ ≡ ψ› shows ‹⊢□ □φ ≡ □ψ›
2097  using "RM:3[prem]" assms by blast
2098
2099lemmas RE = "RM:3"
2100
2101AOT_theorem "RM:4[prem]": assumes ‹Γ ⊢□ φ ≡ ψ› shows ‹□Γ ⊢□ ◇φ ≡ ◇ψ›
2102proof -
2103  AOT_have ‹Γ ⊢□ φ → ψ› and ‹Γ ⊢□ ψ → φ› using assms "≡E" "→I" by metis+
2104  AOT_hence ‹□Γ ⊢□ ◇φ → ◇ψ› and ‹□Γ ⊢□ ◇ψ → ◇φ› using "RM:2[prem]" by metis+
2105  AOT_thus ‹□Γ ⊢□ ◇φ ≡ ◇ψ› by (simp add: "≡I")
2106qed
2107
2108AOT_theorem "RM:4": assumes ‹⊢□ φ ≡ ψ› shows ‹⊢□ ◇φ ≡ ◇ψ›
2109  using "RM:4[prem]" assms by blast
2110
2111lemmas "RE◇" = "RM:4"
2112
2113AOT_theorem "KBasic:1": ‹□φ → □(ψ → φ)›
2114  by (simp add: RM "pl:1"[axiom_inst])
2115
2116AOT_theorem "KBasic:2": ‹□¬φ → □(φ → ψ)›
2117  by (simp add: RM "useful-tautologies:3")
2118
2119AOT_theorem "KBasic:3": ‹□(φ & ψ) ≡ (□φ & □ψ)›
2120proof (rule "≡I"; rule "→I")
2121  AOT_assume ‹□(φ & ψ)›
2122  AOT_thus ‹□φ & □ψ›
2123    by (meson RM "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2124next
2125  AOT_have ‹□φ → □(ψ → (φ & ψ))› by (simp add: "RM:1" Adjunction)
2126  AOT_hence ‹□φ → (□ψ → □(φ & ψ))›  by (metis "Hypothetical Syllogism" "qml:1"[axiom_inst])
2127  moreover AOT_assume ‹□φ & □ψ›
2128  ultimately AOT_show ‹□(φ & ψ)›
2129    using "→E" "&E" by blast
2130qed
2131
2132AOT_theorem "KBasic:4": ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2133proof -
2134  AOT_have θ: ‹□((φ → ψ) & (ψ → φ)) ≡ (□(φ → ψ) & □(ψ → φ))›
2135    by (fact "KBasic:3")
2136  AOT_modally_strict {
2137    AOT_have ‹(φ ≡ ψ) ≡ ((φ → ψ) & (ψ → φ))›
2138      by (fact "conventions:3"[THEN "≡Df"])
2139  }
2140  AOT_hence ξ: ‹□(φ ≡ ψ) ≡ □((φ → ψ) & (ψ → φ))›
2141    by (rule RE)
2142  with ξ and θ AOT_show ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2143    using "≡E"(5) by blast
2144qed
2145
2146AOT_theorem "KBasic:5": ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2147proof -
2148  AOT_have ‹□(φ → ψ) → (□φ → □ψ)›
2149    by (fact "qml:1"[axiom_inst])
2150  moreover AOT_have ‹□(ψ → φ) → (□ψ → □φ)›
2151    by (fact "qml:1"[axiom_inst])
2152  ultimately AOT_have ‹(□(φ → ψ) & □(ψ → φ)) → ((□φ → □ψ) & (□ψ → □φ))›
2153    by (metis "&I" MP "Double Composition")
2154  moreover AOT_have ‹((□φ → □ψ) & (□ψ → □φ)) → (□φ ≡ □ψ)›
2155    using "conventions:3"[THEN "≡dfI"] "→I" by blast
2156  ultimately AOT_show ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2157    by (metis "Hypothetical Syllogism")
2158qed
2159
2160AOT_theorem "KBasic:6": ‹□(φ≡ ψ) → (□φ ≡ □ψ)›
2161  using "KBasic:4" "KBasic:5" "deduction-theorem" "≡E"(1) "vdash-properties:10" by blast
2162AOT_theorem "KBasic:7": ‹((□φ & □ψ) ∨ (□¬φ & □¬ψ)) → □(φ ≡ ψ)›
2163proof (rule "→I"; drule "∨E"(1); (rule "→I")?)
2164  AOT_assume ‹□φ & □ψ›
2165  AOT_hence ‹□φ› and ‹□ψ› using "&E" by blast+
2166  AOT_hence ‹□(φ → ψ)› and ‹□(ψ → φ)› using "KBasic:1" "→E" by blast+
2167  AOT_hence ‹□(φ → ψ) & □(ψ → φ)› using "&I" by blast
2168  AOT_thus ‹□(φ ≡ ψ)›  by (metis "KBasic:4" "≡E"(2))
2169next
2170  AOT_assume ‹□¬φ & □¬ψ›
2171  AOT_hence 0: ‹□(¬φ & ¬ψ)› using "KBasic:3"[THEN "≡E"(2)] by blast
2172  AOT_modally_strict {
2173    AOT_have ‹(¬φ & ¬ψ) → (φ ≡ ψ)›
2174      by (metis "&E"(1) "&E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
2175  }
2176  AOT_hence ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2177    by (rule RM)
2178  AOT_thus ‹□(φ ≡ ψ)› using 0 "→E" by blast
2179qed(auto)
2180
2181AOT_theorem "KBasic:8": ‹□(φ & ψ) → □(φ ≡ ψ)›
2182  by (meson "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I")
2183AOT_theorem "KBasic:9": ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2184  by (metis "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I" "raa-cor:4")
2185AOT_theorem "KBasic:10": ‹□φ ≡ □¬¬φ›
2186  by (simp add: "RM:3" "oth-class-taut:3:b")
2187AOT_theorem "KBasic:11": ‹¬□φ ≡ ◇¬φ›
2188proof (rule "≡I"; rule "→I")
2189  AOT_show ‹◇¬φ› if ‹¬□φ›
2190    using that "≡dfI" "conventions:5" "KBasic:10" "≡E"(3) by blast
2191next
2192  AOT_show ‹¬□φ› if ‹◇¬φ›
2193    using "≡dfE" "conventions:5" "KBasic:10" "≡E"(4) that by blast
2194qed
2195AOT_theorem "KBasic:12": ‹□φ ≡ ¬◇¬φ›
2196proof (rule "≡I"; rule "→I")
2197  AOT_show ‹¬◇¬φ› if ‹□φ›
2198    using "¬¬I" "KBasic:11" "≡E"(3) that by blast
2199next
2200  AOT_show ‹□φ› if ‹¬◇¬φ›
2201  using "KBasic:11" "≡E"(1) "reductio-aa:1" that by blast
2202qed
2203AOT_theorem "KBasic:13": ‹□(φ → ψ) → (◇φ → ◇ψ)›
2204proof -
2205  AOT_have ‹φ → ψ ⊢□ φ → ψ› by blast
2206  AOT_hence ‹□(φ → ψ) ⊢□ ◇φ → ◇ψ›
2207    using "RM:2[prem]" by blast
2208  AOT_thus ‹□(φ → ψ) → (◇φ → ◇ψ)› using "→I" by blast
2209qed
2210lemmas "K◇" = "KBasic:13"
2211AOT_theorem "KBasic:14": ‹◇□φ ≡ ¬□◇¬φ›
2212  by (meson "RE◇" "KBasic:11" "KBasic:12" "≡E"(6) "oth-class-taut:3:a")
2213AOT_theorem "KBasic:15": ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2214proof -
2215  AOT_modally_strict {
2216    AOT_have ‹φ → (φ ∨ ψ)› and ‹ψ → (φ ∨ ψ)›
2217      by (auto simp: "Disjunction Addition"(1) "Disjunction Addition"(2))
2218  }
2219  AOT_hence ‹□φ → □(φ ∨ ψ)› and ‹□ψ → □(φ ∨ ψ)›
2220    using RM by blast+
2221  AOT_thus ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2222    by (metis "∨E"(1) "deduction-theorem")
2223qed
2224
2225AOT_theorem "KBasic:16": ‹(□φ & ◇ψ) → ◇(φ & ψ)›
2226  by (meson "KBasic:13" "RM:1" Adjunction "Hypothetical Syllogism" Importation "vdash-properties:6")
2227
2228AOT_theorem "rule-sub-lem:1:a":
2229  assumes ‹⊢□ □(ψ ≡ χ)›
2230  shows ‹⊢□ ¬ψ ≡ ¬χ›
2231  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2232        "≡E"(1) "oth-class-taut:4:b" by blast
2233
2234AOT_theorem "rule-sub-lem:1:b":
2235  assumes ‹⊢□ □(ψ ≡ χ)›
2236  shows ‹⊢□ (ψ → Θ) ≡ (χ → Θ)›
2237  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2238  using "oth-class-taut:4:c" "vdash-properties:6" by blast
2239
2240AOT_theorem "rule-sub-lem:1:c":
2241  assumes ‹⊢□ □(ψ ≡ χ)›
2242  shows ‹⊢□ (Θ → ψ) ≡ (Θ → χ)›
2243  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2244  using "oth-class-taut:4:d" "vdash-properties:6" by blast
2245
2246AOT_theorem "rule-sub-lem:1:d":
2247  assumes ‹for arbitrary α: ⊢□ □(ψ{α} ≡ χ{α})›
2248  shows ‹⊢□ ∀α ψ{α} ≡ ∀α χ{α}›
2249proof -
2250  AOT_modally_strict {
2251    AOT_have ‹∀α (ψ{α} ≡ χ{α})›
2252      using "qml:2"[axiom_inst, THEN "→E", OF assms] "∀I" by fast
2253    AOT_hence 0: ‹ψ{α} ≡ χ{α}› for α using "∀E" by blast
2254    AOT_show ‹∀α ψ{α} ≡ ∀α χ{α}›
2255    proof (rule "≡I"; rule "→I")
2256      AOT_assume ‹∀α ψ{α}›
2257      AOT_hence ‹ψ{α}› for α using "∀E" by blast
2258      AOT_hence ‹χ{α}› for α using 0 "≡E" by blast
2259      AOT_thus ‹∀α χ{α}› by (rule "∀I")
2260    next
2261      AOT_assume ‹∀α χ{α}›
2262      AOT_hence ‹χ{α}› for α using "∀E" by blast
2263      AOT_hence ‹ψ{α}› for α using 0 "≡E" by blast
2264      AOT_thus ‹∀α ψ{α}› by (rule "∀I")
2265    qed
2266  }
2267qed
2268
2269AOT_theorem "rule-sub-lem:1:e":
2270  assumes ‹⊢□ □(ψ ≡ χ)›
2271  shows ‹⊢□ [λ ψ] ≡ [λ χ]›
2272  using "qml:2"[axiom_inst, THEN "→E", OF assms]
2273  using "≡E"(1) "propositions-lemma:6" by blast
2274
2275AOT_theorem "rule-sub-lem:1:f":
2276  assumes ‹⊢□ □(ψ ≡ χ)›
2277  shows ‹⊢□ 𝒜ψ ≡ 𝒜χ›
2278  using "qml:2"[axiom_inst, THEN "→E", OF assms, THEN "RA[2]"]
2279  by (metis "Act-Basic:5" "≡E"(1))
2280
2281AOT_theorem "rule-sub-lem:1:g":
2282  assumes ‹⊢□ □(ψ ≡ χ)›
2283  shows ‹⊢□ □ψ ≡ □χ›
2284  using "KBasic:6" assms "vdash-properties:6" by blast
2285
2286text‹Note that instead of deriving @{text "rule-sub-lem:2"}, @{text "rule-sub-lem:3"}, @{text "rule-sub-lem:4"},
2287     and @{text "rule-sub-nec"}, we construct substitution methods instead.›
2288
2289class AOT_subst =
2290  fixes AOT_subst :: "('a ⇒ 𝗈) ⇒ bool"
2291    and AOT_subst_cond :: "'a ⇒ 'a ⇒ bool"
2292  assumes AOT_subst: "AOT_subst φ ⟹ AOT_subst_cond ψ χ ⟹ [v ⊨ «φ ψ» ≡ «φ χ»]"
2293
2294named_theorems AOT_substI
2295
2296instantiation 𝗈 :: AOT_subst
2297begin
2298
2299inductive AOT_subst_𝗈 where
2300  AOT_subst_𝗈_id[AOT_substI]: "AOT_subst_𝗈 (λφ. φ)"
2301| AOT_subst_𝗈_const[AOT_substI]: "AOT_subst_𝗈 (λφ. ψ)"
2302| AOT_subst_𝗈_not[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «¬Θ{φ}»)"
2303| AOT_subst_𝗈_imp[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 (λ φ. «Θ{φ} → Ξ{φ}»)"
2304| AOT_subst_𝗈_lambda0[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. (AOT_lambda0 (Θ φ)))"
2305| AOT_subst_𝗈_act[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «𝒜Θ{φ}»)"
2306| AOT_subst_𝗈_box[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «□Θ{φ}»)"
2307| AOT_subst_𝗈_by_def[AOT_substI]: "(⋀ ψ . AOT_model_equiv_def (Θ ψ) (Ξ ψ)) ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 Θ"
2308
2309definition AOT_subst_cond_𝗈 where "AOT_subst_cond_𝗈 ≡ λ ψ χ . ∀ v . [v ⊨ ψ ≡ χ]"
2310
2311instance
2312proof
2313  fix ψ χ :: 𝗈 and φ :: ‹𝗈 ⇒ 𝗈›
2314  assume cond: ‹AOT_subst_cond ψ χ›
2315  assume ‹AOT_subst φ›
2316  moreover AOT_have ‹⊢□ ψ ≡ χ› using cond unfolding AOT_subst_cond_𝗈_def by blast
2317  ultimately AOT_show ‹⊢□ φ{ψ} ≡ φ{χ}›
2318  proof (induct arbitrary: ψ χ)
2319    case AOT_subst_𝗈_id
2320    thus ?case using "≡E"(2) "oth-class-taut:4:b" "rule-sub-lem:1:a" by blast
2321  next
2322    case (AOT_subst_𝗈_const ψ)
2323    thus ?case by (simp add: "oth-class-taut:3:a")
2324  next
2325    case (AOT_subst_𝗈_not Θ)
2326    thus ?case by (simp add: RN "rule-sub-lem:1:a")
2327  next
2328    case (AOT_subst_𝗈_imp Θ Ξ)
2329    thus ?case by (meson RN "≡E"(5) "rule-sub-lem:1:b" "rule-sub-lem:1:c")
2330  next
2331    case (AOT_subst_𝗈_lambda0 Θ)
2332    thus ?case by (simp add: RN "rule-sub-lem:1:e")
2333  next
2334    case (AOT_subst_𝗈_act Θ)
2335    thus ?case by (simp add: RN "rule-sub-lem:1:f")
2336  next
2337    case (AOT_subst_𝗈_box Θ)
2338    thus ?case by (simp add: RN "rule-sub-lem:1:g")
2339  next
2340    case (AOT_subst_𝗈_by_def Θ Ξ)
2341    AOT_modally_strict {
2342      AOT_have ‹Ξ{ψ} ≡ Ξ{χ}› using AOT_subst_𝗈_by_def by simp
2343      AOT_thus ‹Θ{ψ} ≡ Θ{χ}›
2344        using "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ ψ] "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ χ]
2345        by (metis "≡E"(6) "oth-class-taut:3:a")
2346    }
2347  qed
2348qed
2349end
2350
2351instantiation "fun" :: (AOT_Term_id_2, AOT_subst) AOT_subst
2352begin
2353
2354definition AOT_subst_cond_fun :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b) ⇒ bool" where
2355  "AOT_subst_cond_fun ≡ λ φ ψ . ∀ α . AOT_subst_cond (φ (AOT_term_of_var α)) (ψ (AOT_term_of_var α))"
2356
2357inductive AOT_subst_fun :: "(('a ⇒ 'b) ⇒ 𝗈) ⇒ bool" where
2358  AOT_subst_fun_const[AOT_substI]: "AOT_subst_fun (λφ. ψ)"
2359| AOT_subst_fun_id[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. Ψ (φ (AOT_term_of_var x)))"
2360| AOT_subst_fun_all[AOT_substI]: "AOT_subst Ψ ⟹ (⋀ α . AOT_subst_fun (Θ (AOT_term_of_var α))) ⟹ AOT_subst_fun (λφ :: 'a ⇒ 'b. Ψ «∀α «Θ (α::'a) φ»»)"
2361| AOT_subst_fun_not[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. «¬«Ψ φ»»)"
2362| AOT_subst_fun_imp[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst Θ ⟹ AOT_subst_fun (λφ. ««Ψ φ» → «Θ φ»»)"
2363| AOT_subst_fun_lambda0[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. (AOT_lambda0 (Θ φ)))"
2364| AOT_subst_fun_act[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «𝒜«Θ φ»»)"
2365| AOT_subst_fun_box[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «□«Θ φ»»)"
2366| AOT_subst_fun_def[AOT_substI]: "(⋀ φ . AOT_model_equiv_def (Θ φ) (Ψ φ)) ⟹ AOT_subst_fun Ψ ⟹ AOT_subst_fun Θ"
2367
2368instance proof
2369  fix ψ χ :: "'a ⇒ 'b" and φ :: ‹('a ⇒ 'b) ⇒ 𝗈›
2370  assume ‹AOT_subst φ›
2371  moreover assume cond: ‹AOT_subst_cond ψ χ›
2372  ultimately AOT_show ‹⊢□ «φ ψ» ≡ «φ χ»›
2373  proof(induct)
2374    case (AOT_subst_fun_const ψ)
2375    then show ?case by (simp add: "oth-class-taut:3:a")
2376  next
2377  case (AOT_subst_fun_id Ψ x)
2378  then show ?case by (simp add: AOT_subst AOT_subst_cond_fun_def) 
2379  next
2380  case (AOT_subst_fun_all Ψ Θ)
2381  AOT_have ‹⊢□ □(Θ{α, «ψ»} ≡ Θ{α, «χ»})› for α
2382    using AOT_subst_fun_all.hyps(3) AOT_subst_fun_all.prems RN by presburger
2383  thus ?case using AOT_subst[OF AOT_subst_fun_all(1)]
2384    by (simp add: RN "rule-sub-lem:1:d" AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def)
2385  next
2386  case (AOT_subst_fun_not Ψ)
2387  then show ?case by (simp add: RN "rule-sub-lem:1:a")
2388  next
2389  case (AOT_subst_fun_imp Ψ Θ)
2390  then show ?case 
2391    unfolding AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def
2392    by (meson "≡E"(5) "oth-class-taut:4:c" "oth-class-taut:4:d" "vdash-properties:6")
2393  next
2394  case (AOT_subst_fun_lambda0 Θ)
2395  then show ?case by (simp add: RN "rule-sub-lem:1:e")
2396  next
2397  case (AOT_subst_fun_act Θ)
2398  then show ?case by (simp add: RN "rule-sub-lem:1:f")
2399  next
2400  case (AOT_subst_fun_box Θ)
2401  then show ?case by (simp add: RN "rule-sub-lem:1:g")
2402  next
2403  case (AOT_subst_fun_def Θ Ψ)
2404  then show ?case
2405    by (meson "df-rules-formulas[3]" "df-rules-formulas[4]" "≡I" "≡E"(5))
2406  qed
2407qed
2408end
2409
2410method_setup AOT_defI =
2411‹Scan.lift (Scan.succeed (fn ctxt => (Method.CONTEXT_METHOD (fn thms => (Context_Tactic.CONTEXT_SUBGOAL (fn (trm,int) => 
2412Context_Tactic.CONTEXT_TACTIC (
2413let
2414fun findHeadConst (Const x) = SOME x
2415  | findHeadConst (A $ B) = findHeadConst A
2416  | findHeadConst _ = NONE
2417fun findDef (Const (const_name‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2418  | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2419  | findDef (Abs (a,b,c)) = findDef c
2420  | findDef _ = NONE
2421val const_opt = (findDef trm)
2422val defs = case const_opt of SOME const => List.filter (fn thm => let
2423    val concl = Thm.concl_of thm
2424    val thmconst = (findDef concl)
2425    in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2426    | _ => []
2427in
2428resolve_tac ctxt defs 1
2429end
2430)) 1)))))›
2431‹Resolve AOT definitions›
2432
2433
2434method AOT_subst_intro_helper = ((rule AOT_substI
2435      | AOT_defI
2436      | (simp only: AOT_subst_cond_𝗈_def AOT_subst_cond_fun_def; ((rule allI)+)?)))
2437
2438method_setup AOT_subst = ‹
2439Scan.option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
2440Scan.lift (Args.embedded_inner_syntax -- Args.embedded_inner_syntax
2441) -- Scan.option (Scan.lift (Args.$$$ "bound" -- Args.colon) |-- Scan.repeat1 (Scan.lift (Args.embedded_inner_syntax)))
2442>> (fn ((reversed,(raw_p,raw_q)),raw_bounds) => (fn ctxt =>
2443(Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _, prems = prems, asms = asms, concl = concl, schematics = _} =>
2444let
2445val thms = prems
2446val ctxt' = ctxt
2447val ctxt = Context_Position.set_visible false ctxt
2448val raw_bounds = case raw_bounds of SOME bounds => bounds | _ => [] 
2449val bounds = (map (fn x => Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt x))) raw_bounds
2450
2451val p = AOT_read_term @{nonterminal φ'} ctxt raw_p
2452val p = Syntax.check_term ctxt p
2453val p = fold (fn bound => fn p => let in Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,p)) end) bounds p
2454val p = Syntax.check_term ctxt p
2455val p_ty = Term.type_of p
2456val pat = @{const Trueprop} $ (@{const AOT_model_valid_in} $ Var (("w",0), @{typ w}) $ (Var (("φ",0), Type (type_name‹fun›, [p_ty, @{typ 𝗈}])) $ p))
2457val univ = Unify.matchers (Context.Proof ctxt) [(pat, Thm.term_of concl)]
2458val univ = hd (Seq.list_of univ) (* TODO: choose? try all? filter? *)
2459val phi = the (Envir.lookup univ (("φ",0), Type (type_name‹fun›, [p_ty, @{typ 𝗈}])))
2460
2461val q = AOT_read_term @{nonterminal φ'} ctxt raw_q
2462val q = Syntax.check_term ctxt q
2463val q = fold (fn bound => fn q => let in Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,q)) end) bounds q
2464val q = Syntax.check_term ctxt q
2465
2466(* Reparse to report bounds as fixes. *)
2467val ctxt = Context_Position.restore_visible ctxt' ctxt
2468val ctxt' = ctxt
2469fun unsource str = fst (Input.source_content (Syntax.read_input str))
2470val (_,ctxt') = Proof_Context.add_fixes (map (fn str => (Binding.make (unsource str, Position.none), NONE, Mixfix.NoSyn)) raw_bounds) ctxt'
2471val _ = (map (fn x => Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt' x))) raw_bounds
2472val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_p
2473val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_q
2474
2475val abs = phi
2476val abs = HOLogic.mk_Trueprop (@{const AOT_subst(_)} $ abs)
2477val abs = Syntax.check_term ctxt abs
2478
2479val substThm = Goal.prove ctxt [] [] abs (fn {context=ctxt, prems=prems} =>
2480      REPEAT (SUBGOAL (fn (trm,int) => let
2481          fun findHeadConst (Const x) = SOME x
2482            | findHeadConst (A $ B) = findHeadConst A
2483            | findHeadConst _ = NONE
2484          fun findDef (Const (const_name‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2485            | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2486            | findDef (Abs (a,b,c)) = findDef c
2487            | findDef _ = NONE
2488          val const_opt = (findDef trm)
2489          val defs = case const_opt of SOME const => List.filter (fn thm => let
2490              val concl = Thm.concl_of thm
2491              val thmconst = (findDef concl)
2492              in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2493              | _ => []
2494          val tac = case defs of [] => safe_step_tac (ctxt addSIs @{thms AOT_substI}) 1
2495                    | _ => resolve_tac ctxt defs 1
2496        in tac end) 1)
2497  )
2498val substThm = substThm RS @{thm AOT_subst}
2499val abs = Thm.cterm_of ctxt abs
2500val substThm = case reversed of NONE =>
2501  let
2502  val substThm = Drule.instantiate_normalize ([],[((("ψ", 0), p_ty), Thm.cterm_of ctxt p),
2503          ((("χ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
2504  val substThm = substThm RS @{thm "≡E"(2)}
2505  in substThm end
2506| _ =>   let
2507  val substThm = Drule.instantiate_normalize ([],[((("χ", 0), p_ty), Thm.cterm_of ctxt p),
2508          ((("ψ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
2509  val substThm = substThm RS @{thm "≡E"(1)}
2510  in substThm end
2511
2512in
2513resolve_tac ctxt [substThm] 1
2514THEN simp_tac (ctxt addsimps [@{thm AOT_subst_cond_𝗈_def}, @{thm AOT_subst_cond_fun_def}]) 1
2515THEN (REPEAT (resolve_tac ctxt [@{thm allI}] 1))
2516THEN (TRY (resolve_tac ctxt thms 1))
2517end
2518) ctxt 1))))
2519›
2520
2521method AOT_subst_old for ψ::"'a::AOT_subst" and χ::"'a::AOT_subst" =
2522    (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2523      ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2524       ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2525       ; (AOT_subst_intro_helper+)?››)
2526
2527method AOT_subst_old_rev for χ::"'a::AOT_subst" and ψ::"'a::AOT_subst" =
2528    (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2529      ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2530       ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=χ and χ=ψ, THEN "≡E"(1)]
2531       ; (AOT_subst_intro_helper+)?››)
2532
2533method AOT_subst_manual for φ::"'a::AOT_subst ⇒ 𝗈" =
2534    (rule AOT_subst[where φ=φ, THEN "≡E"(2)]; (AOT_subst_intro_helper+)?)
2535
2536method AOT_subst_manual_rev for φ::"'a::AOT_subst ⇒ 𝗈" =
2537    (rule AOT_subst[where φ=φ, THEN "≡E"(1)]; (AOT_subst_intro_helper+)?)
2538
2539method AOT_subst_using uses subst =
2540    (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2541       match conclusion in "[v ⊨ «φ ψ»]" for φ v ⇒ ‹
2542         rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2543         ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2544
2545method AOT_subst_using_rev uses subst =
2546    (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2547      match conclusion in "[v ⊨ «φ χ»]" for φ v ⇒ ‹
2548        rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(1)]
2549        ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2550
2551AOT_theorem "rule-sub-remark:1[1]": assumes ‹⊢□ A!x ≡ ¬◇E!x› and ‹¬A!x› shows ‹¬¬◇E!x›
2552  by (AOT_subst (reverse) ‹¬◇E!x› ‹A!x›)
2553     (auto simp: assms) 
2554
2555AOT_theorem "rule-sub-remark:1[2]": assumes ‹⊢□ A!x ≡ ¬◇E!x› and  ‹¬¬◇E!x› shows ‹¬A!x›
2556  by (AOT_subst ‹A!x› ‹¬◇E!x›)
2557     (auto simp: assms)
2558
2559AOT_theorem "rule-sub-remark:2[1]":
2560  assumes ‹⊢□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy› shows ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)›
2561  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2562
2563AOT_theorem "rule-sub-remark:2[2]":
2564  assumes ‹⊢□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)› shows ‹p → [R]xy›
2565  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2566
2567AOT_theorem "rule-sub-remark:3[1]":
2568  assumes ‹for arbitrary x: ⊢□ A!x ≡ ¬◇E!x›
2569      and ‹∃x A!x›
2570    shows ‹∃x ¬◇E!x›
2571  by (AOT_subst (reverse) ‹¬◇E!x› ‹A!x› bound: x)
2572     (auto simp: assms)
2573
2574AOT_theorem "rule-sub-remark:3[2]":
2575  assumes ‹for arbitrary x: ⊢□ A!x ≡ ¬◇E!x›
2576      and ‹∃x ¬◇E!x›
2577    shows ‹∃x A!x›
2578  by (AOT_subst ‹A!x› ‹¬◇E!x› bound: x)
2579     (auto simp: assms)
2580
2581AOT_theorem "rule-sub-remark:4[1]":
2582  assumes ‹⊢□ ¬¬[P]x ≡ [P]x› and ‹𝒜¬¬[P]x› shows ‹𝒜[P]x›
2583  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2584
2585AOT_theorem "rule-sub-remark:4[2]":
2586  assumes ‹⊢□ ¬¬[P]x ≡ [P]x› and ‹𝒜[P]x› shows ‹𝒜¬¬[P]x›
2587  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2588
2589AOT_theorem "rule-sub-remark:5[1]":
2590  assumes ‹⊢□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(φ → ψ)› shows ‹□(¬ψ → ¬φ)›
2591  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2592
2593AOT_theorem "rule-sub-remark:5[2]":
2594  assumes ‹⊢□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(¬ψ → ¬φ)› shows ‹□(φ → ψ)› 
2595  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2596
2597AOT_theorem "rule-sub-remark:6[1]":
2598  assumes ‹⊢□ ψ ≡ χ› and ‹□(φ → ψ)› shows ‹□(φ → χ)› 
2599  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2600
2601AOT_theorem "rule-sub-remark:6[2]":
2602  assumes ‹⊢□ ψ ≡ χ› and ‹□(φ → χ)› shows ‹□(φ → ψ)›
2603  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2604
2605AOT_theorem "rule-sub-remark:7[1]":
2606  assumes ‹⊢□ φ ≡ ¬¬φ› and ‹□(φ → φ)› shows ‹□(¬¬φ → φ)› 
2607  by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2608
2609AOT_theorem "rule-sub-remark:7[2]":
2610  assumes ‹⊢□ φ ≡ ¬¬φ› and ‹□(¬¬φ → φ)› shows  ‹□(φ → φ)›
2611  by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2612
2613AOT_theorem "KBasic2:1": ‹□¬φ ≡ ¬◇φ›
2614  by (meson "conventions:5" "contraposition:2" "Hypothetical Syllogism" "df-rules-formulas[3]"
2615            "df-rules-formulas[4]" "≡I" "useful-tautologies:1")
2616
2617AOT_theorem "KBasic2:2": ‹◇(φ ∨ ψ) ≡ (◇φ ∨ ◇ψ)›
2618proof -
2619  AOT_have ‹◇(φ ∨ ψ) ≡ ◇¬(¬φ & ¬ψ)›
2620    by (simp add: "RE◇" "oth-class-taut:5:b")
2621  also AOT_have ‹… ≡ ¬□(¬φ & ¬ψ)›
2622    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2623  also AOT_have ‹… ≡ ¬(□¬φ & □¬ψ)›
2624    using "KBasic:3" "≡E"(1) "oth-class-taut:4:b" by blast
2625  also AOT_have ‹… ≡ ¬(¬◇φ & ¬◇ψ)›
2626    using "KBasic2:1"
2627    by (AOT_subst ‹□¬φ› ‹¬◇φ›; AOT_subst ‹□¬ψ› ‹¬◇ψ›; auto simp: "oth-class-taut:3:a")
2628  also AOT_have ‹… ≡ ¬¬(◇φ ∨ ◇ψ)›
2629    using "≡E"(6) "oth-class-taut:3:b" "oth-class-taut:5:b" by blast
2630  also AOT_have ‹… ≡ ◇φ ∨ ◇ψ›
2631    by (simp add: "≡I" "useful-tautologies:1" "useful-tautologies:2")
2632  finally show ?thesis .
2633qed
2634
2635AOT_theorem "KBasic2:3": ‹◇(φ & ψ) → (◇φ & ◇ψ)›
2636  by (metis "RM◇" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "deduction-theorem" "modus-tollens:1" "reductio-aa:1")
2637
2638AOT_theorem "KBasic2:4": ‹◇(φ → ψ) ≡ (□φ → ◇ψ)›
2639proof -
2640  AOT_have ‹◇(φ → ψ) ≡ ◇(¬φ ∨ ψ)›
2641    by (AOT_subst ‹φ → ψ› ‹¬φ ∨ ψ›)
2642       (auto simp: "oth-class-taut:1:c" "oth-class-taut:3:a")
2643  also AOT_have ‹... ≡ ◇¬φ ∨ ◇ψ›
2644    by (simp add: "KBasic2:2")
2645  also AOT_have ‹... ≡ ¬□φ ∨ ◇ψ›
2646    by (AOT_subst ‹¬□φ› ‹◇¬φ›)
2647       (auto simp: "KBasic:11" "oth-class-taut:3:a")
2648  also AOT_have ‹... ≡ □φ → ◇ψ›
2649    using "≡E"(6) "oth-class-taut:1:c" "oth-class-taut:3:a" by blast
2650  finally show ?thesis .
2651qed
2652
2653AOT_theorem "KBasic2:5": ‹◇◇φ ≡ ¬□□¬φ›
2654  using "conventions:5"[THEN "≡Df"]
2655  by (AOT_subst ‹◇φ› ‹¬□¬φ›; AOT_subst ‹◇¬□¬φ› ‹¬□¬¬□¬φ›; AOT_subst (reverse) ‹¬¬□¬φ› ‹□¬φ›)
2656     (auto simp: "oth-class-taut:3:b" "oth-class-taut:3:a")
2657
2658
2659AOT_theorem "KBasic2:6": ‹□(φ ∨ ψ) → (□φ ∨ ◇ψ)›
2660proof(rule "→I"; rule "raa-cor:1")
2661  AOT_assume ‹□(φ ∨ ψ)›
2662  AOT_hence ‹□(¬φ → ψ)›
2663    using "conventions:2"[THEN "≡Df"]
2664    by (AOT_subst (reverse) ‹¬φ → ψ› ‹φ ∨ ψ›) simp
2665  AOT_hence 1: ‹◇¬φ → ◇ψ› using "KBasic:13" "vdash-properties:10" by blast
2666  AOT_assume ‹¬(□φ ∨ ◇ψ)›
2667  AOT_hence ‹¬□φ› and ‹¬◇ψ› using "&E" "≡E"(1) "oth-class-taut:5:d" by blast+
2668  AOT_thus ‹◇ψ & ¬◇ψ› using "&I"(1) 1[THEN "→E"] "KBasic:11" "≡E"(4) "raa-cor:3" by blast
2669qed
2670
2671AOT_theorem "KBasic2:7": ‹(□(φ ∨ ψ) & ◇¬φ) → ◇ψ›
2672proof(rule "→I"; frule "&E"(1); drule "&E"(2))
2673  AOT_assume ‹□(φ ∨ ψ)›
2674  AOT_hence 1: ‹□φ ∨ ◇ψ›
2675    using "KBasic2:6" "∨I"(2) "∨E"(1) by blast
2676  AOT_assume ‹◇¬φ›
2677  AOT_hence ‹¬□φ› using "KBasic:11" "≡E"(2) by blast
2678  AOT_thus ‹◇ψ› using 1 "∨E"(2) by blast
2679qed
2680
2681AOT_theorem "T-S5-fund:1": ‹φ → ◇φ›
2682  by (meson "≡dfI" "conventions:5" "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" "qml:2"[axiom_inst])
2683lemmas "T◇" = "T-S5-fund:1"
2684
2685AOT_theorem "T-S5-fund:2": ‹◇□φ → □φ›
2686proof(rule "→I")
2687  AOT_assume ‹◇□φ›
2688  AOT_hence ‹¬□◇¬φ›
2689    using "KBasic:14" "≡E"(4) "raa-cor:3" by blast
2690  moreover AOT_have ‹◇¬φ → □◇¬φ›
2691    by (fact "qml:3"[axiom_inst])
2692  ultimately AOT_have ‹¬◇¬φ›
2693    using "modus-tollens:1" by blast
2694  AOT_thus ‹□φ› using "KBasic:12" "≡E"(2) by blast
2695qed
2696lemmas "5◇" = "T-S5-fund:2"
2697
2698(* Also interestingly none of these have proofs in PLM. *)
2699AOT_theorem "Act-Sub:1": ‹𝒜φ ≡ ¬𝒜¬φ›
2700  by (AOT_subst ‹𝒜¬φ› ‹¬𝒜φ›)
2701     (auto simp: "logic-actual-nec:1"[axiom_inst] "oth-class-taut:3:b")
2702
2703AOT_theorem "Act-Sub:2": ‹◇φ ≡ 𝒜◇φ›
2704  using "conventions:5"[THEN "≡Df"]
2705  by (AOT_subst ‹◇φ› ‹¬□¬φ›)
2706     (metis "deduction-theorem" "≡I" "≡E"(1) "≡E"(2) "≡E"(3)
2707            "logic-actual-nec:1"[axiom_inst] "qml-act:2"[axiom_inst])
2708
2709AOT_theorem "Act-Sub:3": ‹𝒜φ → ◇φ›
2710  using "conventions:5"[THEN "≡Df"]
2711  by (AOT_subst ‹◇φ› ‹¬□¬φ›)
2712     (metis "Act-Sub:1" "deduction-theorem" "≡E"(4) "nec-imp-act" "reductio-aa:2" "→E")
2713
2714
2715AOT_theorem "Act-Sub:4": ‹𝒜φ ≡ ◇𝒜φ›
2716proof (rule "≡I"; rule "→I")
2717  AOT_assume ‹𝒜φ›
2718  AOT_thus ‹◇𝒜φ› using "T◇" "vdash-properties:10" by blast
2719next
2720  AOT_assume ‹◇𝒜φ›
2721  AOT_hence ‹¬□¬𝒜φ›
2722    using "≡dfE" "conventions:5" by blast
2723  AOT_hence ‹¬□𝒜¬φ›
2724    by (AOT_subst ‹𝒜¬φ› ‹¬𝒜φ›)
2725       (simp add: "logic-actual-nec:1"[axiom_inst])
2726  AOT_thus ‹𝒜φ›
2727      using "Act-Basic:1" "Act-Basic:6" "∨E"(3) "≡E"(4) "reductio-aa:1" by blast
2728qed
2729
2730AOT_theorem "Act-Sub:5": ‹◇𝒜φ → 𝒜◇φ›
2731  by (metis "Act-Sub:2" "Act-Sub:3" "Act-Sub:4" "deduction-theorem" "≡E"(1) "≡E"(2) "vdash-properties:6")
2732
2733AOT_theorem "S5Basic:1": ‹◇φ ≡ □◇φ›
2734  by (simp add: "≡I" "qml:2" "qml:3" "vdash-properties:1[2]")
2735
2736AOT_theorem "S5Basic:2": ‹□φ ≡ ◇□φ›
2737  by (simp add: "T◇" "5◇" "≡I")
2738
2739AOT_theorem "S5Basic:3": ‹φ → □◇φ›
2740  using "T◇" "Hypothetical Syllogism" "qml:3" "vdash-properties:1[2]" by blast
2741lemmas "B" = "S5Basic:3"
2742
2743AOT_theorem "S5Basic:4": ‹◇□φ → φ›
2744  using "5◇" "Hypothetical Syllogism" "qml:2" "vdash-properties:1[2]" by blast
2745lemmas "B◇" = "S5Basic:4"
2746
2747AOT_theorem "S5Basic:5": ‹□φ → □□φ›
2748  using "RM:1" "B" "5◇" "Hypothetical Syllogism" by blast
2749lemmas "4" = "S5Basic:5"
2750
2751AOT_theorem "S5Basic:6": ‹□φ ≡ □□φ›
2752  by (simp add: "4" "≡I" "qml:2"[axiom_inst])
2753
2754AOT_theorem "S5Basic:7": ‹◇◇φ → ◇φ›
2755  using "conventions:5"[THEN "≡Df"] "oth-class-taut:3:b"
2756  by (AOT_subst ‹◇◇φ› ‹¬□¬◇φ›;
2757      AOT_subst ‹◇φ› ‹¬□¬φ›;
2758      AOT_subst (reverse) ‹¬¬□¬φ› ‹□¬φ›;
2759      AOT_subst (reverse) ‹□□¬φ› ‹□¬φ›)
2760     (auto simp: "S5Basic:6" "if-p-then-p")
2761
2762lemmas "4◇" = "S5Basic:7"
2763
2764AOT_theorem "S5Basic:8": ‹◇◇φ ≡ ◇φ›
2765  by (simp add: "4◇" "T◇" "≡I")
2766
2767AOT_theorem "S5Basic:9": ‹□(φ ∨ □ψ) ≡ (□φ ∨ □ψ)›
2768  apply (rule "≡I"; rule "→I")
2769  using "KBasic2:6" "5◇" "∨I"(3) "if-p-then-p" "vdash-properties:10" apply blast
2770  by (meson "KBasic:15" "4" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "con-dis-taut:7"
2771            "intro-elim:1" "Commutativity of ∨")
2772
2773AOT_theorem "S5Basic:10": ‹□(φ ∨ ◇ψ) ≡ (□φ ∨ ◇ψ)›
2774(* Note: nicely this proof is entirely sledgehammer generated *)
2775proof(rule "≡I"; rule "→I")
2776  AOT_assume ‹□(φ ∨ ◇ψ)›
2777  AOT_hence ‹□φ ∨ ◇◇ψ›
2778    by (meson "KBasic2:6" "∨I"(2) "∨E"(1))
2779  AOT_thus ‹□φ ∨ ◇ψ›
2780    by (meson "B◇" "4" "4◇" "T◇" "∨I"(3))
2781next
2782  AOT_assume ‹□φ ∨ ◇ψ›
2783  AOT_hence ‹□φ ∨ □◇ψ›
2784    by (meson "S5Basic:1" "B◇" "S5Basic:6" "T◇" "5◇" "∨I"(3) "intro-elim:1")
2785  AOT_thus ‹□(φ ∨ ◇ψ)›
2786    by (meson "KBasic:15" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
2787qed
2788
2789AOT_theorem "S5Basic:11": ‹◇(φ & ◇ψ) ≡ (◇φ & ◇ψ)›
2790proof -
2791  AOT_have ‹◇(φ & ◇ψ) ≡ ◇¬(¬φ ∨ ¬◇ψ)›
2792    by (AOT_subst ‹φ & ◇ψ› ‹¬(¬φ ∨ ¬◇ψ)›)
2793       (auto simp: "oth-class-taut:5:a" "oth-class-taut:3:a")
2794  also AOT_have ‹… ≡ ◇¬(¬φ ∨ □¬ψ)›
2795    by (AOT_subst ‹□¬ψ› ‹¬◇ψ›)
2796       (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2797  also AOT_have ‹… ≡ ¬□(¬φ ∨ □¬ψ)›
2798    using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2799  also AOT_have ‹… ≡ ¬(□¬φ ∨ □¬ψ)›
2800    using "S5Basic:9" "≡E"(1) "oth-class-taut:4:b" by blast
2801  also AOT_have ‹… ≡ ¬(¬◇φ ∨ ¬◇ψ)›
2802    using "KBasic2:1"
2803    by (AOT_subst ‹□¬φ› ‹¬◇φ›; AOT_subst ‹□¬ψ› ‹¬◇ψ›)
2804       (auto simp:  "oth-class-taut:3:a")
2805  also AOT_have ‹… ≡ ◇φ & ◇ψ›
2806    using "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:a" by blast
2807  finally show ?thesis .
2808qed
2809
2810AOT_theorem "S5Basic:12": ‹◇(φ & □ψ) ≡ (◇φ & □ψ)›
2811proof (rule "≡I"; rule "→I")
2812  AOT_assume ‹◇(φ & □ψ)›
2813  AOT_hence ‹◇φ & ◇□ψ›
2814    using "KBasic2:3" "vdash-properties:6" by blast
2815  AOT_thus ‹◇φ & □ψ›
2816    using "5◇" "&I" "&E"(1) "&E"(2) "vdash-properties:6" by blast
2817next
2818  AOT_assume ‹◇φ & □ψ›
2819  moreover AOT_have ‹(□□ψ & ◇φ) → ◇(φ & □ψ)›
2820    by (AOT_subst ‹φ & □ψ› ‹□ψ & φ›)
2821       (auto simp: "Commutativity of &" "KBasic:16")
2822  ultimately AOT_show ‹◇(φ & □ψ)›
2823    by (metis "4" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2824qed
2825
2826
2827AOT_theorem "S5Basic:13": ‹□(φ → □ψ) ≡ □(◇φ → ψ)›
2828proof (rule "≡I")
2829  AOT_modally_strict {
2830    AOT_have ‹□(φ → □ψ) → (◇φ → ψ)›
2831      by (meson "KBasic:13" "B◇" "Hypothetical Syllogism" "deduction-theorem")
2832  }
2833  AOT_hence ‹□□(φ → □ψ) → □(◇φ → ψ)›
2834    by (rule RM)
2835  AOT_thus  ‹□(φ → □ψ) → □(◇φ → ψ)›
2836    using "4" "Hypothetical Syllogism" by blast
2837next
2838  AOT_modally_strict {
2839    AOT_have ‹□(◇φ → ψ) → (φ → □ψ)›
2840      by (meson "B" "Hypothetical Syllogism" "deduction-theorem" "qml:1" "vdash-properties:1[2]")
2841  }
2842  AOT_hence  ‹□□(◇φ → ψ) → □(φ → □ψ)›
2843    by (rule RM)
2844  AOT_thus ‹□(◇φ → ψ) → □(φ → □ψ)›
2845    using "4" "Hypothetical Syllogism" by blast
2846qed
2847
2848AOT_theorem "derived-S5-rules:1":
2849  assumes ‹Γ ⊢□ ◇φ → ψ› shows ‹□Γ ⊢□ φ → □ψ›
2850proof -
2851  AOT_have ‹□Γ ⊢□ □◇φ → □ψ›
2852    using assms by (rule "RM:1[prem]")
2853  AOT_thus ‹□Γ ⊢□ φ → □ψ›
2854    using "B" "Hypothetical Syllogism" by blast
2855qed
2856
2857AOT_theorem "derived-S5-rules:2":
2858  assumes ‹Γ ⊢□ φ → □ψ› shows ‹□Γ ⊢□ ◇φ → ψ›
2859proof -
2860  AOT_have ‹□Γ ⊢□ ◇φ → ◇□ψ›
2861    using assms by (rule "RM:2[prem]")
2862  AOT_thus ‹□Γ ⊢□ ◇φ → ψ›
2863    using "B◇" "Hypothetical Syllogism" by blast
2864qed
2865
2866AOT_theorem "BFs:1": ‹∀α □φ{α} → □∀α φ{α}›
2867proof -
2868  AOT_modally_strict {
2869    AOT_modally_strict {
2870      AOT_have ‹∀α □φ{α} → □φ{α}› for α by (fact AOT)
2871    }
2872    AOT_hence ‹◇∀α □φ{α} → ◇□φ{α}› for α by (rule "RM◇")
2873    AOT_hence ‹◇∀α □φ{α} → ∀α φ{α}›
2874      using "B◇" "∀I" "→E" "→I" by metis
2875  }
2876  thus ?thesis using "derived-S5-rules:1" by blast
2877qed
2878lemmas "BF" = "BFs:1"
2879
2880AOT_theorem "BFs:2": ‹□∀α φ{α} → ∀α □φ{α}›
2881proof -
2882  AOT_have ‹□∀α φ{α} → □φ{α}› for α using RM "cqt-orig:3" by metis
2883  thus ?thesis using  "cqt-orig:2"[THEN "→E"] "∀I" by metis
2884qed
2885lemmas "CBF" = "BFs:2"
2886
2887AOT_theorem "BFs:3": ‹◇∃α φ{α} → ∃α ◇φ{α}›
2888proof(rule "→I")
2889  AOT_modally_strict {
2890    AOT_have ‹□∀α ¬φ{α} ≡ ∀α □¬φ{α}›
2891      using BF CBF "≡I" by blast
2892  } note θ = this
2893
2894  AOT_assume ‹◇∃α φ{α}›
2895  AOT_hence ‹¬□¬(∃α φ{α})›
2896    using "≡dfE" "conventions:5" by blast
2897  AOT_hence ‹¬□∀α ¬φ{α}›
2898    apply (AOT_subst ‹∀α ¬φ{α}› ‹¬(∃α φ{α})›)
2899    using "≡dfI" "conventions:3" "conventions:4" "&I" "contraposition:2" "cqt-further:4"
2900          "df-rules-formulas[1]" "vdash-properties:1[2]" by blast
2901  AOT_hence ‹¬∀α □¬φ{α}›
2902    apply (AOT_subst (reverse) ‹∀α □¬φ{α}› ‹□∀α ¬φ{α}›)
2903    using θ by blast
2904  AOT_hence ‹¬∀α ¬¬□¬φ{α}›
2905    apply - apply (AOT_subst_old_rev "λ τ. «□¬φ{τ}»"  "λ τ. «¬¬□¬φ{τ}»")
2906    by (simp add: "oth-class-taut:3:b")
2907  AOT_hence 0: ‹∃α ¬□¬φ{α}›
2908    by (rule "conventions:4"[THEN "≡dfI"])
2909  AOT_show ‹∃α ◇φ{α}›
2910    apply (AOT_subst_old "λ τ . «◇φ{τ}»" "λ τ . «¬□¬φ{τ}»")
2911     apply (simp add: "conventions:5" "≡Df")
2912    using 0 by blast
2913qed
2914lemmas "BF◇" = "BFs:3"
2915
2916AOT_theorem "BFs:4": ‹∃α ◇φ{α} → ◇∃α φ{α}›
2917proof(rule "→I")
2918  AOT_assume ‹∃α ◇φ{α}›
2919  AOT_hence ‹¬∀α ¬◇φ{α}›
2920    using "conventions:4"[THEN "≡dfE"] by blast
2921  AOT_hence ‹¬∀α □¬φ{α}›
2922    apply - apply (AOT_subst_old "λ τ . «□¬φ{τ}»" "λ τ . «¬◇φ{τ}»")
2923    by (simp add: "KBasic2:1")
2924  moreover AOT_have ‹∀α □¬φ{α} ≡ □∀α ¬φ{α}›
2925    using "≡I" "BF" "CBF" by metis
2926  ultimately AOT_have 1: ‹¬□∀α ¬φ{α}›
2927    using "≡E"(3) by blast
2928  AOT_show ‹◇∃α φ{α}›
2929    apply (rule "conventions:5"[THEN "≡dfI"])
2930    apply (AOT_subst ‹∃α φ{α}› ‹¬∀α ¬φ{α}›)
2931     apply (simp add: "conventions:4" "≡Df")
2932    apply (AOT_subst ‹¬¬∀α ¬φ{α}› ‹∀α ¬φ{α}›)
2933    by (auto simp: 1 "≡I" "useful-tautologies:1" "useful-tautologies:2")
2934qed
2935lemmas "CBF◇" = "BFs:4"
2936
2937AOT_theorem "sign-S5-thm:1": ‹∃α □φ{α} → □∃α φ{α}›
2938proof(rule "→I")
2939  AOT_assume ‹∃α □φ{α}›
2940  then AOT_obtain α where ‹□φ{α}› using "∃E" by metis
2941  moreover AOT_have ‹□α↓›
2942    by (simp add: "ex:1:a" "rule-ui:2[const_var]" RN)
2943  moreover AOT_have ‹□φ{τ}, □τ↓ ⊢□ □∃α φ{α}› for τ
2944  proof -
2945    AOT_have ‹φ{τ}, τ↓ ⊢□ ∃α φ{α}› using "existential:1" by blast
2946    AOT_thus ‹□φ{τ}, □τ↓ ⊢□ □∃α φ{α}›
2947      using "RN[prem]"[where Γ="{φ τ, «τ↓»}", simplified] by blast
2948  qed
2949  ultimately AOT_show ‹□∃α φ{α}› by blast
2950qed
2951lemmas Buridan = "sign-S5-thm:1"
2952
2953AOT_theorem "sign-S5-thm:2": ‹◇∀α φ{α} → ∀α ◇φ{α}›
2954proof -
2955  AOT_have ‹∀α (◇∀α φ{α} → ◇φ{α})›
2956    by (simp add: "RM◇" "cqt-orig:3" "∀I")
2957  AOT_thus ‹◇∀α φ{α} → ∀α ◇φ{α}›
2958    using "∀E"(4) "∀I" "→E" "→I" by metis
2959qed
2960lemmas "Buridan◇" = "sign-S5-thm:2"
2961
2962AOT_theorem "sign-S5-thm:3": ‹◇∃α (φ{α} & ψ{α}) → ◇(∃α φ{α} & ∃α ψ{α})›
2963  apply (rule "RM:2")
2964  by (metis (no_types, lifting) "instantiation" "&I" "&E"(1)
2965                                "&E"(2) "deduction-theorem" "existential:2[const_var]")
2966
2967AOT_theorem "sign-S5-thm:4": ‹◇∃α (φ{α} & ψ{α}) → ◇∃α φ{α}›
2968  apply (rule "RM:2")
2969  by (meson "instantiation" "&E"(1) "deduction-theorem" "existential:2[const_var]")
2970
2971AOT_theorem "sign-S5-thm:5": ‹(□∀α (φ{α} → ψ{α}) & □∀α (ψ{α} → χ{α})) → □∀α (φ{α} → χ{α})›
2972proof -
2973  {
2974    fix φ' ψ' χ'
2975    AOT_assume ‹⊢□ φ' & ψ' → χ'›
2976    AOT_hence ‹□φ' & □ψ' → □χ'›
2977      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2978      using "&E" "&I" "→E" "→I" by metis
2979  } note R = this
2980  show ?thesis by (rule R; fact AOT)
2981qed
2982
2983AOT_theorem "sign-S5-thm:6": ‹(□∀α (φ{α} ≡ ψ{α}) & □∀α(ψ{α} ≡ χ{α})) → □∀α(φ{α} ≡ χ{α})›
2984proof -
2985  {
2986    fix φ' ψ' χ'
2987    AOT_assume ‹⊢□ φ' & ψ' → χ'›
2988    AOT_hence ‹□φ' & □ψ' → □χ'›
2989      using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2990      using "&E" "&I" "→E" "→I" by metis
2991  } note R = this
2992  show ?thesis by (rule R; fact AOT)
2993qed
2994
2995AOT_theorem "exist-nec2:1": ‹◇τ↓ → τ↓›
2996  using "B◇" "RM◇" "Hypothetical Syllogism" "exist-nec" by blast
2997
2998AOT_theorem "exists-nec2:2": ‹◇τ↓ ≡ □τ↓›
2999  by (meson "Act-Sub:3" "Hypothetical Syllogism" "exist-nec" "exist-nec2:1" "≡I" "nec-imp-act")
3000
3001AOT_theorem "exists-nec2:3": ‹¬τ↓ → □¬τ↓›
3002  using "KBasic2:1" "deduction-theorem" "exist-nec2:1" "≡E"(2) "modus-tollens:1" by blast
3003
3004AOT_theorem "exists-nec2:4": ‹◇¬τ↓ ≡ □¬τ↓›
3005  by (metis "Act-Sub:3" "KBasic:12" "deduction-theorem" "exist-nec" "exists-nec2:3" "≡I" "≡E"(4) "nec-imp-act" "reductio-aa:1")
3006
3007AOT_theorem "id-nec2:1": ‹◇α = β → α = β›
3008  using "B◇" "RM◇" "Hypothetical Syllogism" "id-nec:1" by blast
3009
3010AOT_theorem "id-nec2:2": ‹α ≠ β → □α ≠ β›
3011  apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
3012  using "KBasic2:1" "deduction-theorem" "id-nec2:1" "≡E"(2) "modus-tollens:1" by blast
3013
3014AOT_theorem "id-nec2:3": ‹◇α ≠ β → α ≠ β›
3015  apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
3016  by (metis "KBasic:11" "deduction-theorem" "id-nec:2" "≡E"(3) "reductio-aa:2" "vdash-properties:6")
3017
3018AOT_theorem "id-nec2:4": ‹◇α = β → □α = β›
3019  using "Hypothetical Syllogism" "id-nec2:1" "id-nec:1" by blast
3020
3021AOT_theorem "id-nec2:5": ‹◇α ≠ β → □α ≠ β›
3022  using "id-nec2:3" "id-nec2:2" "→I" "→E" by metis
3023
3024AOT_theorem "sc-eq-box-box:1": ‹□(φ → □φ) ≡ (◇φ → □φ)›
3025  apply (rule "≡I"; rule "→I")
3026  using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:10" apply blast
3027  by (metis "KBasic2:1" "KBasic:1" "KBasic:2" "S5Basic:13" "≡E"(2) "raa-cor:5" "vdash-properties:6")
3028
3029AOT_theorem "sc-eq-box-box:2": ‹(□(φ → □φ) ∨ (◇φ → □φ)) → (◇φ ≡ □φ)›
3030  by (metis "Act-Sub:3" "KBasic:13" "5◇" "∨E"(2) "deduction-theorem" "≡I" "nec-imp-act" "raa-cor:2" "vdash-properties:10")
3031
3032AOT_theorem "sc-eq-box-box:3": ‹□(φ → □φ) → (¬□φ ≡ □¬φ)›
3033proof (rule "→I"; rule "≡I"; rule "→I")
3034  AOT_assume ‹□(φ → □φ)›
3035  AOT_hence ‹◇φ → □φ› using "sc-eq-box-box:1" "≡E" by blast
3036  moreover AOT_assume ‹¬□φ›
3037  ultimately AOT_have ‹¬◇φ›
3038    using "modus-tollens:1" by blast
3039  AOT_thus ‹□¬φ›
3040    using "KBasic2:1" "≡E"(2) by blast
3041next
3042  AOT_assume ‹□(φ → □φ)›
3043  moreover AOT_assume ‹□¬φ›
3044  ultimately AOT_show ‹¬□φ›
3045    using "modus-tollens:1" "qml:2" "vdash-properties:10" "vdash-properties:1[2]" by blast
3046qed
3047
3048AOT_theorem "sc-eq-box-box:4": ‹(□(φ → □φ) & □(ψ → □ψ)) → ((□φ ≡ □ψ) → □(φ ≡ ψ))›
3049proof(rule "→I"; rule "→I")
3050  AOT_assume θ: ‹□(φ → □φ) & □(ψ → □ψ)›
3051  AOT_assume ξ: ‹□φ ≡ □ψ›
3052  AOT_hence ‹(□φ & □ψ) ∨ (¬□φ & ¬□ψ)›
3053    using "≡E"(4) "oth-class-taut:4:g" "raa-cor:3" by blast
3054  moreover {
3055    AOT_assume ‹□φ & □ψ›
3056    AOT_hence ‹□(φ ≡ ψ)›
3057      using "KBasic:3" "KBasic:8" "≡E"(2) "vdash-properties:10" by blast
3058  }
3059  moreover {
3060    AOT_assume ‹¬□φ & ¬□ψ›
3061    moreover AOT_have ‹¬□φ ≡ □¬φ› and ‹¬□ψ ≡ □¬ψ›
3062      using θ "Conjunction Simplification"(1) "Conjunction Simplification"(2) "sc-eq-box-box:3" "vdash-properties:10" by metis+
3063    ultimately AOT_have ‹□¬φ & □¬ψ›
3064      by (metis "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "≡E"(4) "modus-tollens:1" "raa-cor:3")
3065    AOT_hence ‹□(φ ≡ ψ)›
3066      using "KBasic:3" "KBasic:9" "≡E"(2) "vdash-properties:10" by blast
3067  }
3068  ultimately AOT_show ‹□(φ ≡ ψ)›
3069    using "∨E"(2) "reductio-aa:1" by blast
3070qed
3071
3072AOT_theorem "sc-eq-box-box:5": ‹(□(φ → □φ) & □(ψ → □ψ)) → □((φ ≡ ψ) → □(φ ≡ ψ))›
3073proof (rule "→I")
3074  AOT_assume ‹(□(φ → □φ) & □(ψ → □ψ))›
3075  AOT_hence ‹□(□(φ → □φ) & □(ψ → □ψ))›
3076    using 4[THEN "→E"] "&E" "&I" "KBasic:3" "≡E"(2) by metis
3077  moreover AOT_have ‹□(□(φ → □φ) & □(ψ → □ψ)) → □((φ ≡ ψ) → □(φ ≡ ψ))›
3078  proof (rule RM; rule "→I"; rule "→I")
3079    AOT_modally_strict {
3080      AOT_assume A: ‹(□(φ → □φ) & □(ψ → □ψ))›
3081      AOT_hence ‹φ → □φ› and ‹ψ → □ψ›
3082        using "&E" "qml:2"[axiom_inst] "→E" by blast+
3083      moreover AOT_assume ‹φ ≡ ψ›
3084      ultimately AOT_have ‹□φ ≡ □ψ›
3085        using "→E" "qml:2"[axiom_inst] "≡E" "≡I" by meson
3086      moreover AOT_have ‹(□φ ≡ □ψ) → □(φ ≡ ψ)›
3087        using A "sc-eq-box-box:4" "→E" by blast
3088      ultimately AOT_show ‹□(φ ≡ ψ)› using "→E" by blast
3089    }
3090  qed
3091  ultimately AOT_show ‹□((φ ≡ ψ) → □(φ ≡ ψ))› using "→E" by blast
3092qed
3093
3094AOT_theorem "sc-eq-box-box:6": ‹□(φ → □φ) → ((φ → □ψ) → □(φ → ψ))›
3095proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3096  AOT_assume ‹¬□(φ → ψ)›
3097  AOT_hence ‹◇¬(φ → ψ)› by (metis "KBasic:11" "≡E"(1))
3098  AOT_hence ‹◇(φ & ¬ψ)›
3099    by (AOT_subst ‹φ & ¬ψ› ‹¬(φ → ψ)›)
3100       (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3101  AOT_hence ‹◇φ› and 2: ‹◇¬ψ› using "KBasic2:3"[THEN "→E"] "&E" by blast+
3102  moreover AOT_assume ‹□(φ → □φ)›
3103  ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3104  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3105  moreover AOT_assume ‹φ → □ψ›
3106  ultimately AOT_have ‹□ψ› using "→E" by blast
3107  moreover AOT_have ‹¬□ψ› using 2 "KBasic:12" "¬¬I" "intro-elim:3:d" by blast
3108  ultimately AOT_show ‹□ψ & ¬□ψ› using "&I" by blast
3109qed
3110
3111AOT_theorem "sc-eq-box-box:7": ‹□(φ → □φ) → ((φ → 𝒜ψ) → 𝒜(φ → ψ))›
3112proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3113  AOT_assume ‹¬𝒜(φ → ψ)›
3114  AOT_hence ‹𝒜¬(φ → ψ)› by (metis "Act-Basic:1" "∨E"(2))
3115  AOT_hence ‹𝒜(φ & ¬ψ)›
3116    by (AOT_subst ‹φ & ¬ψ› ‹¬(φ → ψ)›)
3117       (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3118  AOT_hence ‹𝒜φ› and 2: ‹𝒜¬ψ› using "Act-Basic:2"[THEN "≡E"(1)] "&E" by blast+
3119  AOT_hence ‹◇φ› by (metis "Act-Sub:3" "→E")
3120  moreover AOT_assume ‹□(φ → □φ)›
3121  ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3122  AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3123  moreover AOT_assume ‹φ → 𝒜ψ›
3124  ultimately AOT_have ‹𝒜ψ› using "→E" by blast
3125  moreover AOT_have ‹¬𝒜ψ› using 2 by (meson "Act-Sub:1" "≡E"(4) "raa-cor:3")
3126  ultimately AOT_show ‹𝒜ψ & ¬𝒜ψ› using "&I" by blast
3127qed
3128
3129AOT_theorem "sc-eq-fur:1": ‹◇𝒜φ ≡ □𝒜φ›
3130  using "Act-Basic:6" "Act-Sub:4" "≡E"(6) by blast
3131
3132AOT_theorem "sc-eq-fur:2": ‹□(φ → □φ) → (𝒜φ ≡ φ)›
3133  by (metis "B◇" "Act-Sub:3" "KBasic:13" "T◇" "Hypothetical Syllogism" "deduction-theorem" "≡I" "nec-imp-act")
3134
3135AOT_theorem "sc-eq-fur:3": ‹□∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ιx φ{x}↓)›
3136proof (rule "→I"; rule "→I")
3137  AOT_assume ‹□∀x (φ{x} → □φ{x})›
3138  AOT_hence A: ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3139  AOT_assume ‹∃!x φ{x}›
3140  then AOT_obtain a where a_def: ‹φ{a} & ∀y (φ{y} → y = a)›
3141    using "∃E"[rotated 1, OF "uniqueness:1"[THEN "≡dfE"]] by blast
3142  moreover AOT_have ‹□φ{a}› using calculation A "∀E"(2) "qml:2"[axiom_inst] "→E" "&E"(1) by blast
3143  AOT_hence ‹𝒜φ{a}› using "nec-imp-act" "vdash-properties:6" by blast
3144  moreover AOT_have ‹∀y (𝒜φ{y} → y = a)›
3145  proof (rule "∀I"; rule "→I")
3146    fix b
3147    AOT_assume ‹𝒜φ{b}›
3148    AOT_hence ‹◇φ{b}›
3149      using "Act-Sub:3" "vdash-properties:6" by blast
3150    moreover {
3151      AOT_have ‹□(φ{b} → □φ{b})›
3152        using A "∀E"(2) by blast
3153      AOT_hence ‹◇φ{b} → □φ{b}›
3154        using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:6" by blast
3155    }
3156    ultimately AOT_have ‹□φ{b}› using "→E" by blast
3157    AOT_hence ‹φ{b}› using "qml:2"[axiom_inst] "→E" by blast
3158    AOT_thus ‹b = a›
3159      using a_def[THEN "&E"(2)] "∀E"(2) "→E" by blast
3160  qed
3161  ultimately AOT_have ‹𝒜φ{a} & ∀y (𝒜φ{y} → y = a)›
3162    using "&I" by blast
3163  AOT_hence ‹∃x (𝒜φ{x} & ∀y (𝒜φ{y} → y = x))› using "∃I" by fast
3164  AOT_hence ‹∃!x 𝒜φ{x}› using "uniqueness:1"[THEN "≡dfI"] by fast
3165  AOT_thus ‹ιx φ{x}↓›
3166    using "actual-desc:1"[THEN "≡E"(2)] by blast
3167qed
3168
3169AOT_theorem "sc-eq-fur:4": ‹□∀x (φ{x} → □φ{x}) → (x = ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x)))›
3170proof (rule "→I")
3171  AOT_assume ‹□∀x (φ{x} → □φ{x})›
3172  AOT_hence ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3173  AOT_hence A: ‹𝒜φ{α} ≡ φ{α}› for α using "sc-eq-fur:2" "∀E" "→E" by fast
3174  AOT_show ‹x = ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x))›
3175  proof (rule "≡I"; rule "→I")
3176    AOT_assume ‹x = ιx φ{x}›
3177    AOT_hence B: ‹𝒜φ{x} & ∀z (𝒜φ{z} → z = x)›
3178      using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
3179    AOT_show ‹φ{x} & ∀z (φ{z} → z = x)›
3180    proof (rule "&I"; (rule "∀I"; rule "→I")?)
3181      AOT_show ‹φ{x}› using A B[THEN "&E"(1)] "≡E"(1) by blast
3182    next
3183      AOT_show ‹z = x› if ‹φ{z}› for z
3184        using that B[THEN "&E"(2)] "∀E"(2) "→E" A[THEN "≡E"(2)] by blast
3185    qed
3186  next
3187    AOT_assume B: ‹φ{x} & ∀z (φ{z} → z = x)›
3188    AOT_have ‹𝒜φ{x} & ∀z (𝒜φ{z} → z = x)›
3189    proof(rule "&I"; (rule "∀I"; rule "→I")?)
3190      AOT_show ‹𝒜φ{x}› using B[THEN "&E"(1)] A[THEN "≡E"(2)] by blast
3191    next
3192      AOT_show ‹b = x› if ‹𝒜φ{b}› for b
3193        using that A[THEN "≡E"(1)] B[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] by blast
3194    qed
3195    AOT_thus ‹x = ιx φ{x}›
3196      using "nec-hintikka-scheme"[THEN "≡E"(2)] by blast
3197  qed
3198qed
3199
3200AOT_theorem "id-act:1": ‹α = β ≡ 𝒜α = β›
3201  by (meson "Act-Sub:3" "Hypothetical Syllogism" "id-nec2:1" "id-nec:2" "≡I" "nec-imp-act")
3202
3203AOT_theorem "id-act:2": ‹α ≠ β ≡ 𝒜α ≠ β›
3204proof (AOT_subst ‹α ≠ β› ‹¬(α = β)›)
3205  AOT_modally_strict {
3206    AOT_show ‹α ≠ β ≡ ¬(α = β)›
3207      by (simp add: "=-infix" "≡Df")
3208  }
3209next
3210  AOT_show ‹¬(α = β) ≡ 𝒜¬(α = β)›
3211  proof (safe intro!: "≡I" "→I")
3212    AOT_assume ‹¬α = β›
3213    AOT_hence ‹¬𝒜α = β› using "id-act:1" "≡E"(3) by blast
3214    AOT_thus ‹𝒜¬α = β›
3215      using "¬¬E" "Act-Sub:1" "≡E"(3) by blast
3216  next
3217    AOT_assume ‹𝒜¬α = β›
3218    AOT_hence ‹¬𝒜α = β›
3219      using "¬¬I" "Act-Sub:1" "≡E"(4) by blast
3220    AOT_thus ‹¬α = β›
3221      using "id-act:1" "≡E"(4) by blast
3222  qed
3223qed
3224
3225AOT_theorem "A-Exists:1": ‹𝒜∃!α φ{α} ≡ ∃!α 𝒜φ{α}›
3226proof -
3227  AOT_have ‹𝒜∃!α φ{α} ≡ 𝒜∃α∀β (φ{β} ≡ β = α)›
3228    by (AOT_subst_using subst: "uniqueness:2")
3229       (simp add: "oth-class-taut:3:a")
3230  also AOT_have ‹… ≡ ∃α 𝒜∀β (φ{β} ≡ β = α)›
3231    by (simp add: "Act-Basic:10")
3232  also AOT_have ‹… ≡ ∃α∀β 𝒜(φ{β} ≡ β = α)›
3233    by (AOT_subst_old "λ τ . «𝒜∀β (φ{β} ≡ β = τ)»" "λ τ . «∀β 𝒜(φ{β} ≡ β = τ)»")
3234       (auto simp: "logic-actual-nec:3" "vdash-properties:1[2]" "oth-class-taut:3:a")
3235  also AOT_have ‹… ≡ ∃α∀β (𝒜φ{β} ≡ 𝒜β = α)›
3236    by (AOT_subst_old_rev "λ τ τ' . «𝒜(φ{τ'} ≡ τ' = τ)»" "λ τ τ'. «𝒜φ{τ'} ≡ 𝒜τ' = τ»")
3237       (auto simp: "Act-Basic:5" "cqt-further:7")
3238  also AOT_have ‹… ≡ ∃α∀β (𝒜φ{β} ≡ β = α)›
3239    apply (AOT_subst_old "λ τ τ' :: 'a . «𝒜τ' = τ»" "λ τ τ'. «τ' = τ»")
3240     apply (meson "id-act:1" "≡E"(6) "oth-class-taut:3:a")
3241    by (simp add: "cqt-further:7")
3242  also AOT_have ‹... ≡ ∃!α 𝒜φ{α}›
3243    using "uniqueness:2" "Commutativity of ≡"[THEN "≡E"(1)] by fast
3244  finally show ?thesis .
3245qed
3246
3247AOT_theorem "A-Exists:2": ‹ιx φ{x}↓ ≡ 𝒜∃!x φ{x}›
3248  by (AOT_subst_using subst: "A-Exists:1")
3249     (simp add: "actual-desc:1")
3250
3251AOT_theorem "id-act-desc:1": ‹ιx (x = y)↓›
3252proof(rule "existence:1"[THEN "≡dfI"]; rule "∃I")
3253  AOT_show ‹[λx E!x → E!x]ιx (x = y)›
3254  proof (rule "russell-axiom[exe,1].nec-russell-axiom"[THEN "≡E"(2)]; rule "∃I"; (rule "&I")+)
3255    AOT_show ‹𝒜y = y› by (simp add: "RA[2]" "id-eq:1")
3256  next
3257    AOT_show ‹∀z (𝒜z = y → z = y)›
3258      apply (rule "∀I")
3259      using "id-act:1"[THEN "≡E"(2)] "→I" by blast
3260  next
3261    AOT_show ‹[λx E!x → E!x]y›
3262    proof (rule "lambda-predicates:2"[axiom_inst, THEN "→E", THEN "≡E"(2)])
3263      AOT_show ‹[λx E!x → E!x]↓›
3264        by "cqt:2[lambda]"
3265    next
3266      AOT_show ‹E!y → E!y› 
3267        by (simp add: "if-p-then-p")
3268    qed
3269  qed
3270next
3271  AOT_show ‹[λx E!x → E!x]↓›
3272    by "cqt:2[lambda]"
3273qed
3274
3275AOT_theorem "id-act-desc:2": ‹y = ιx (x = y)›
3276  by (rule descriptions[axiom_inst, THEN "≡E"(2)]; rule "∀I"; rule "id-act:1"[symmetric])
3277
3278AOT_theorem "pre-en-eq:1[1]": ‹x1[F] → □x1[F]›
3279  by (simp add: encoding "vdash-properties:1[2]")
3280
3281AOT_theorem "pre-en-eq:1[2]": ‹x1x2[F] → □x1x2[F]›
3282proof (rule "→I")
3283  AOT_assume ‹x1x2[F]›
3284  AOT_hence ‹x1[λy [F]yx2]› and ‹x2[λy [F]x1y]›
3285    using "nary-encoding[2]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3286  moreover AOT_have ‹[λy [F]yx2]↓› by "cqt:2[lambda]"
3287  moreover AOT_have ‹[λy [F]x1y]↓› by "cqt:2[lambda]"
3288  ultimately AOT_have ‹□x1[λy [F]yx2]› and ‹□x2[λy [F]x1y]›
3289    using encoding[axiom_inst, unvarify F] "→E" "&I" by blast+
3290  note A = this
3291  AOT_hence ‹□(x1[λy [F]yx2] & x2[λy [F]x1y])›
3292    using "KBasic:3"[THEN "≡E"(2)] "&I" by blast
3293  AOT_thus ‹□x1x2[F]›
3294    by (rule "nary-encoding[2]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3295qed
3296
3297AOT_theorem "pre-en-eq:1[3]": ‹x1x2x3[F] → □x1x2x3[F]›
3298proof (rule "→I")
3299  AOT_assume ‹x1x2x3[F]›
3300  AOT_hence ‹x1[λy [F]yx2x3]› and ‹x2[λy [F]x1yx3]› and ‹x3[λy [F]x1x2y]›
3301    using "nary-encoding[3]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3302  moreover AOT_have ‹[λy [F]yx2x3]↓› by "cqt:2[lambda]"
3303  moreover AOT_have ‹[λy [F]x1yx3]↓› by "cqt:2[lambda]"
3304  moreover AOT_have ‹[λy [F]x1x2y]↓› by "cqt:2[lambda]"
3305  ultimately AOT_have ‹□x1[λy [F]yx2x3]› and ‹□x2[λy [F]x1yx3]› and ‹□x3[λy [F]x1x2y]›
3306    using encoding[axiom_inst, unvarify F] "→E" by blast+
3307  note A = this
3308  AOT_have B: ‹□(x1[λy [F]yx2x3] & x2[λy [F]x1yx3] & x3[λy [F]x1x2y])›
3309    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3310  AOT_thus ‹□x1x2x3[F]›
3311    by (rule "nary-encoding[3]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3312qed
3313
3314AOT_theorem "pre-en-eq:1[4]": ‹x1x2x3x4[F] → □x1x2x3x4[F]›
3315proof (rule "→I")
3316  AOT_assume ‹x1x2x3x4[F]›
3317  AOT_hence ‹x1[λy [F]yx2x3x4]› and ‹x2[λy [F]x1yx3x4]› and ‹x3[λy [F]x1x2yx4]› and  ‹x4[λy [F]x1x2x3y]›
3318    using "nary-encoding[4]"[axiom_inst, THEN "≡E"(1)] "&E" by metis+
3319  moreover AOT_have ‹[λy [F]yx2x3x4]↓› by "cqt:2[lambda]"
3320  moreover AOT_have ‹[λy [F]x1yx3x4]↓› by "cqt:2[lambda]"
3321  moreover AOT_have ‹[λy [F]x1x2yx4]↓› by "cqt:2[lambda]"
3322  moreover AOT_have ‹[λy [F]x1x2x3y]↓› by "cqt:2[lambda]"
3323  ultimately AOT_have ‹□x1[λy [F]yx2x3x4]› and ‹□x2[λy [F]x1yx3x4]› and ‹□x3[λy [F]x1x2yx4]› and ‹□x4[λy [F]x1x2x3y]›
3324    using "→E" encoding[axiom_inst, unvarify F] by blast+
3325  note A = this
3326  AOT_have B: ‹□(x1[λy [F]yx2x3x4] & x2[λy [F]x1yx3x4] & x3[λy [F]x1x2yx4] & x4[λy [F]x1x2x3y])›
3327    by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3328  AOT_thus ‹□x1x2x3x4[F]›
3329    by (rule "nary-encoding[4]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3330qed
3331
3332AOT_theorem "pre-en-eq:2[1]": ‹¬x1[F] → □¬x1[F]›
3333proof (rule "→I"; rule "raa-cor:1")
3334  AOT_assume ‹¬□¬x1[F]›
3335  AOT_hence ‹◇x1[F]›
3336    by (rule "conventions:5"[THEN "≡dfI"])
3337  AOT_hence ‹x1[F]›
3338    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[1]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3339  moreover AOT_assume ‹¬x1[F]›
3340  ultimately AOT_show ‹x1[F] & ¬x1[F]› by (rule "&I")
3341qed
3342AOT_theorem "pre-en-eq:2[2]": ‹¬x1x2[F] → □¬x1x2[F]›
3343proof (rule "→I"; rule "raa-cor:1")
3344  AOT_assume ‹¬□¬x1x2[F]›
3345  AOT_hence ‹◇x1x2[F]›
3346    by (rule "conventions:5"[THEN "≡dfI"])
3347  AOT_hence ‹x1x2[F]›
3348    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[2]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3349  moreover AOT_assume ‹¬x1x2[F]›
3350  ultimately AOT_show ‹x1x2[F] & ¬x1x2[F]› by (rule "&I")
3351qed
3352
3353AOT_theorem "pre-en-eq:2[3]": ‹¬x1x2x3[F] → □¬x1x2x3[F]›
3354proof (rule "→I"; rule "raa-cor:1")
3355  AOT_assume ‹¬□¬x1x2x3[F]›
3356  AOT_hence ‹◇x1x2x3[F]›
3357    by (rule "conventions:5"[THEN "≡dfI"])
3358  AOT_hence ‹x1x2x3[F]›
3359    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[3]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3360  moreover AOT_assume ‹¬x1x2x3[F]›
3361  ultimately AOT_show ‹x1x2x3[F] & ¬x1x2x3[F]› by (rule "&I")
3362qed
3363
3364AOT_theorem "pre-en-eq:2[4]": ‹¬x1x2x3x4[F] → □¬x1x2x3x4[F]›
3365proof (rule "→I"; rule "raa-cor:1")
3366  AOT_assume ‹¬□¬x1x2x3x4[F]›
3367  AOT_hence ‹◇x1x2x3x4[F]›
3368    by (rule "conventions:5"[THEN "≡dfI"])
3369  AOT_hence ‹x1x2x3x4[F]›
3370    by(rule "S5Basic:13"[THEN "≡E"(1), OF  "pre-en-eq:1[4]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3371  moreover AOT_assume ‹¬x1x2x3x4[F]›
3372  ultimately AOT_show ‹x1x2x3x4[F] & ¬x1x2x3x4[F]› by (rule "&I")
3373qed
3374
3375AOT_theorem "en-eq:1[1]": ‹◇x1[F] ≡ □x1[F]›
3376  using "pre-en-eq:1[1]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3377AOT_theorem "en-eq:1[2]": ‹◇x1x2[F] ≡ □x1x2[F]›
3378  using "pre-en-eq:1[2]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3379AOT_theorem "en-eq:1[3]": ‹◇x1x2x3[F] ≡ □x1x2x3[F]›
3380  using "pre-en-eq:1[3]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3381AOT_theorem "en-eq:1[4]": ‹◇x1x2x3x4[F] ≡ □x1x2x3x4[F]›
3382  using "pre-en-eq:1[4]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3383
3384AOT_theorem "en-eq:2[1]": ‹x1[F] ≡ □x1[F]›
3385  by (simp add: "≡I" "pre-en-eq:1[1]" "qml:2"[axiom_inst])
3386AOT_theorem "en-eq:2[2]": ‹x1x2[F] ≡ □x1x2[F]›
3387  by (simp add: "≡I" "pre-en-eq:1[2]" "qml:2"[axiom_inst])
3388AOT_theorem "en-eq:2[3]": ‹x1x2x3[F] ≡ □x1x2x3[F]›
3389  by (simp add: "≡I" "pre-en-eq:1[3]" "qml:2"[axiom_inst])
3390AOT_theorem "en-eq:2[4]": ‹x1x2x3x4[F] ≡ □x1x2x3x4[F]›
3391  by (simp add: "≡I" "pre-en-eq:1[4]" "qml:2"[axiom_inst])
3392
3393AOT_theorem "en-eq:3[1]": ‹◇x1[F] ≡ x1[F]›
3394  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[1]"] "≡I" by blast
3395AOT_theorem "en-eq:3[2]": ‹◇x1x2[F] ≡ x1x2[F]›
3396  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[2]"] "≡I" by blast
3397AOT_theorem "en-eq:3[3]": ‹◇x1x2x3[F] ≡ x1x2x3[F]›
3398  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[3]"] "≡I" by blast
3399AOT_theorem "en-eq:3[4]": ‹◇x1x2x3x4[F] ≡ x1x2x3x4[F]›
3400  using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[4]"] "≡I" by blast
3401
3402AOT_theorem "en-eq:4[1]": ‹(x1[F] ≡ y1[G]) ≡ (□x1[F] ≡ □y1[G])›
3403  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3404  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[1]" by blast+
3405AOT_theorem "en-eq:4[2]": ‹(x1x2[F] ≡ y1y2[G]) ≡ (□x1x2[F] ≡ □y1y2[G])›
3406  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3407  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[2]" by blast+
3408AOT_theorem "en-eq:4[3]": ‹(x1x2x3[F] ≡ y1y2y3[G]) ≡ (□x1x2x3[F] ≡ □y1y2y3[G])›
3409  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3410  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[3]" by blast+
3411AOT_theorem "en-eq:4[4]": ‹(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ (□x1x2x3x4[F] ≡ □y1y2y3y4[G])›
3412  apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3413  using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[4]" by blast+
3414
3415AOT_theorem "en-eq:5[1]": ‹□(x1[F] ≡ y1[G]) ≡ (□x1[F] ≡ □y1[G])›
3416  apply (rule "≡I"; rule "→I")
3417  using "en-eq:4[1]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3418  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3419        "&I"[OF "pre-en-eq:1[1]"[THEN RN], OF "pre-en-eq:1[1]"[THEN RN]] by blast
3420AOT_theorem "en-eq:5[2]": ‹□(x1x2[F] ≡ y1y2[G]) ≡ (□x1x2[F] ≡ □y1y2[G])›
3421  apply (rule "≡I"; rule "→I")
3422  using "en-eq:4[2]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3423  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3424        "&I"[OF "pre-en-eq:1[2]"[THEN RN], OF "pre-en-eq:1[2]"[THEN RN]] by blast
3425AOT_theorem "en-eq:5[3]": ‹□(x1x2x3[F] ≡ y1y2y3[G]) ≡ (□x1x2x3[F] ≡ □y1y2y3[G])›
3426  apply (rule "≡I"; rule "→I")
3427  using "en-eq:4[3]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3428  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3429        "&I"[OF "pre-en-eq:1[3]"[THEN RN], OF "pre-en-eq:1[3]"[THEN RN]] by blast
3430AOT_theorem "en-eq:5[4]": ‹□(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ (□x1x2x3x4[F] ≡ □y1y2y3y4[G])›
3431  apply (rule "≡I"; rule "→I")
3432  using "en-eq:4[4]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3433  using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3434        "&I"[OF "pre-en-eq:1[4]"[THEN RN], OF "pre-en-eq:1[4]"[THEN RN]] by blast
3435
3436AOT_theorem "en-eq:6[1]": ‹(x1[F] ≡ y1[G]) ≡ □(x1[F] ≡ y1[G])›
3437  using "en-eq:5[1]"[symmetric] "en-eq:4[1]" "≡E"(5) by fast
3438AOT_theorem "en-eq:6[2]": ‹(x1x2[F] ≡ y1y2[G]) ≡ □(x1x2[F] ≡ y1y2[G])›
3439  using "en-eq:5[2]"[symmetric] "en-eq:4[2]" "≡E"(5) by fast
3440AOT_theorem "en-eq:6[3]": ‹(x1x2x3[F] ≡ y1y2y3[G]) ≡ □(x1x2x3[F] ≡ y1y2y3[G])›
3441  using "en-eq:5[3]"[symmetric] "en-eq:4[3]" "≡E"(5) by fast
3442AOT_theorem "en-eq:6[4]": ‹(x1x2x3x4[F] ≡ y1y2y3y4[G]) ≡ □(x1x2x3x4[F] ≡ y1y2y3y4[G])›
3443  using "en-eq:5[4]"[symmetric] "en-eq:4[4]" "≡E"(5) by fast
3444
3445AOT_theorem "en-eq:7[1]": ‹¬x1[F] ≡ □¬x1[F]›
3446  using "pre-en-eq:2[1]" "qml:2"[axiom_inst] "≡I" by blast
3447AOT_theorem "en-eq:7[2]": ‹¬x1x2[F] ≡ □¬x1x2[F]›
3448  using "pre-en-eq:2[2]" "qml:2"[axiom_inst] "≡I" by blast
3449AOT_theorem "en-eq:7[3]": ‹¬x1x2x3[F] ≡ □¬x1x2x3[F]›
3450  using "pre-en-eq:2[3]" "qml:2"[axiom_inst] "≡I" by blast
3451AOT_theorem "en-eq:7[4]": ‹¬x1x2x3x4[F] ≡ □¬x1x2x3x4[F]›
3452  using "pre-en-eq:2[4]" "qml:2"[axiom_inst] "≡I" by blast
3453
3454AOT_theorem "en-eq:8[1]": ‹◇¬x1[F] ≡ ¬x1[F]›
3455  using "en-eq:2[1]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3456AOT_theorem "en-eq:8[2]": ‹◇¬x1x2[F] ≡ ¬x1x2[F]›
3457  using "en-eq:2[2]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3458AOT_theorem "en-eq:8[3]": ‹◇¬x1x2x3[F] ≡ ¬x1x2x3[F]›
3459  using "en-eq:2[3]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3460AOT_theorem "en-eq:8[4]": ‹◇¬x1x2x3x4[F] ≡ ¬x1x2x3x4[F]›
3461  using "en-eq:2[4]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3462
3463AOT_theorem "en-eq:9[1]": ‹◇¬x1[F] ≡ □¬x1[F]›
3464  using "en-eq:7[1]" "en-eq:8[1]" "≡E"(5) by blast
3465AOT_theorem "en-eq:9[2]": ‹◇¬x1x2[F] ≡ □¬x1x2[F]›
3466  using "en-eq:7[2]" "en-eq:8[2]" "≡E"(5) by blast
3467AOT_theorem "en-eq:9[3]": ‹◇¬x1x2x3[F] ≡ □¬x1x2x3[F]›
3468  using "en-eq:7[3]" "en-eq:8[3]" "≡E"(5) by blast
3469AOT_theorem "en-eq:9[4]": ‹◇¬x1x2x3x4[F] ≡ □¬x1x2x3x4[F]›
3470  using "en-eq:7[4]" "en-eq:8[4]" "≡E"(5) by blast
3471
3472AOT_theorem "en-eq:10[1]": ‹𝒜x1[F] ≡ x1[F]›
3473  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[1]" "pre-en-eq:1[1]")
3474AOT_theorem "en-eq:10[2]": ‹𝒜x1x2[F] ≡ x1x2[F]›
3475  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[2]" "pre-en-eq:1[2]")
3476AOT_theorem "en-eq:10[3]": ‹𝒜x1x2x3[F] ≡ x1x2x3[F]›
3477  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[3]" "pre-en-eq:1[3]")
3478AOT_theorem "en-eq:10[4]": ‹𝒜x1x2x3x4[F] ≡ x1x2x3x4[F]›
3479  by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[4]" "pre-en-eq:1[4]")
3480
3481AOT_theorem "oa-facts:1": ‹O!x → □O!x›
3482proof(rule "→I")
3483  AOT_modally_strict {
3484    AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
3485      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3486  } note θ = this
3487  AOT_assume ‹O!x›
3488  AOT_hence ‹[λx ◇E!x]x›
3489    by (rule "=dfE"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3490  AOT_hence ‹◇E!x› using θ[THEN "≡E"(1)] by blast
3491  AOT_hence 0: ‹□◇E!x› using "qml:3"[axiom_inst, THEN "→E"] by blast
3492  AOT_have ‹□[λx ◇E!x]x›
3493    by (AOT_subst_using subst: θ) (simp add: 0)
3494  AOT_thus ‹□O!x›
3495    by (rule "=dfI"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3496qed
3497
3498AOT_theorem "oa-facts:2": ‹A!x → □A!x›
3499proof(rule "→I")
3500  AOT_modally_strict {
3501    AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
3502      by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3503  } note θ = this
3504  AOT_assume ‹A!x›
3505  AOT_hence ‹[λx ¬◇E!x]x›
3506    by (rule "=dfE"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3507  AOT_hence ‹¬◇E!x› using θ[THEN "≡E"(1)] by blast
3508  AOT_hence ‹□¬E!x› using "KBasic2:1"[THEN "≡E"(2)] by blast
3509  AOT_hence 0: ‹□□¬E!x› using "4"[THEN "→E"] by blast
3510  AOT_have 1: ‹□¬◇E!x›
3511    apply (AOT_subst ‹¬◇E!x› ‹□¬E!x›)
3512    using "KBasic2:1"[symmetric] apply blast
3513    using 0 by blast
3514  AOT_have ‹□[λx ¬◇E!x]x›
3515    by (AOT_subst_using subst: θ) (simp add: 1)
3516  AOT_thus ‹□A!x›
3517    by (rule "=dfI"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3518qed
3519
3520AOT_theorem "oa-facts:3": ‹◇O!x → O!x›
3521  using "oa-facts:1" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3522AOT_theorem "oa-facts:4": ‹◇A!x → A!x›
3523  using "oa-facts:2" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3524
3525AOT_theorem "oa-facts:5": ‹◇O!x ≡ □O!x›
3526  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3527
3528AOT_theorem "oa-facts:6": ‹◇A!x ≡ □A!x›
3529  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3530
3531AOT_theorem "oa-facts:7": ‹O!x ≡ 𝒜O!x›
3532  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3533
3534AOT_theorem "oa-facts:8": ‹A!x ≡ 𝒜A!x›
3535  by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3536
3537AOT_theorem "beta-C-meta": ‹[λμ1...μn φ{μ1...μn, ν1...νn}]↓ → ([λμ1...μn φ{μ1...μn, ν1...νn}]ν1...νn ≡ φ{ν1...νn, ν1...νn})›
3538  using "lambda-predicates:2"[axiom_inst] by blast
3539
3540AOT_theorem "beta-C-cor:1": ‹(∀ν1...∀νn([λμ1...μn φ{μ1...μn, ν1...νn}]↓)) → ∀ν1...∀νn ([λμ1...μn φ{μ1...μn, ν1...νn}]ν1...νn ≡ φ{ν1...νn, ν1...νn})›
3541  apply (rule "cqt-basic:14"[where 'a='a, THEN "→E"])
3542  using "beta-C-meta" "∀I" by fast
3543
3544AOT_theorem "beta-C-cor:2": ‹[λμ1...μn φ{μ1...μn}]↓ → ∀ν1...∀νn ([λμ1...μn φ{μ1...μn}]ν1...νn ≡ φ{ν1...νn})›
3545  apply (rule "→I"; rule "∀I")
3546  using "beta-C-meta"[THEN "→E"] by fast
3547
3548(* TODO: syntax + double-check if this is really a faithful representation *)
3549theorem "beta-C-cor:3": assumes ‹⋀ν1νn. AOT_instance_of_cqt_2 (φ (AOT_term_of_var ν1νn))›
3550  shows ‹[v ⊨ ∀ν1...∀νn ([λμ1...μn φ{ν1...νn,μ1...μn}]ν1...νn ≡ φ{ν1...νn,ν1...νn})]›
3551  using "cqt:2[lambda]"[axiom_inst, OF assms] "beta-C-cor:1"[THEN "→E"] "∀I" by fast
3552
3553AOT_theorem "betaC:1:a": ‹[λμ1...μn φ{μ1...μn}]κ1...κn ⊢□ φ{κ1...κn}›
3554proof -
3555  AOT_modally_strict {
3556    AOT_assume ‹[λμ1...μn φ{μ1...μn}]κ1...κn›
3557    moreover AOT_have ‹[λμ1...μn φ{μ1...μn}]↓› and ‹κ1...κn↓›
3558      using calculation "cqt:5:a"[axiom_inst, THEN "→E"] "&E" by blast+
3559    ultimately AOT_show ‹φ{κ1...κn}›
3560      using "beta-C-cor:2"[THEN "→E", THEN "∀E"(1), THEN "≡E"(1)] by blast
3561  }
3562qed
3563
3564AOT_theorem "betaC:1:b": ‹¬φ{κ1...κn} ⊢□ ¬[λμ1...μn φ{μ1...μn}]κ1...κn›
3565  using "betaC:1:a" "raa-cor:3" by blast
3566
3567lemmas "β→C" = "betaC:1:a" "betaC:1:b"
3568
3569AOT_theorem "betaC:2:a": ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, φ{κ1...κn} ⊢□ [λμ1...μn φ{μ1...μn}]κ1...κn›
3570proof -
3571  AOT_modally_strict {
3572    AOT_assume 1: ‹[λμ1...μn φ{μ1...μn}]↓› and 2: ‹κ1...κn↓› and 3: ‹φ{κ1...κn}›
3573    AOT_hence ‹[λμ1...μn φ{μ1...μn}]κ1...κn›
3574      using "beta-C-cor:2"[THEN "→E", OF 1, THEN "∀E"(1), THEN "≡E"(2)] by blast
3575  }
3576  AOT_thus ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, φ{κ1...κn} ⊢□ [λμ1...μn φ{μ1...μn}]κ1...κn›
3577    by blast
3578qed
3579
3580AOT_theorem "betaC:2:b": ‹[λμ1...μn φ{μ1...μn}]↓, κ1...κn↓, ¬[λμ1...μn φ{μ1...μn}]κ1...κn ⊢□ ¬φ{κ1...κn}›
3581  using "betaC:2:a" "raa-cor:3" by blast
3582
3583lemmas "β←C" = "betaC:2:a" "betaC:2:b"
3584
3585AOT_theorem "eta-conversion-lemma1:1": ‹Π↓ → [λx1...xn [Π]x1...xn] = Π›
3586  using "lambda-predicates:3"[axiom_inst] "∀I" "∀E"(1) "→I" by fast
3587
3588AOT_theorem "eta-conversion-lemma1:2": ‹Π↓ → [λν1...νn [Π]ν1...νn] = Π›
3589  using "eta-conversion-lemma1:1". (* TODO: spurious in the embedding *)
3590
3591(* match (τ) in "λa . ?b" ⇒ ‹match (τ') in "λa . ?b" ⇒ ‹fail›› ¦ _ ⇒ ‹ *)
3592
3593text‹Note: not explicitly part of PLM.›
3594AOT_theorem id_sym: assumes ‹τ = τ'› shows ‹τ' = τ›
3595  using "rule=E"[where φ="λ τ' . «τ' = τ»", rotated 1, OF assms]
3596        "=I"(1)[OF "t=t-proper:1"[THEN "→E", OF assms]] by auto
3597declare id_sym[sym]
3598
3599text‹Note: not explicitly part of PLM.›
3600AOT_theorem id_trans: assumes ‹τ = τ'› and ‹τ' = τ''› shows ‹τ = τ''›
3601  using "rule=E" assms by blast
3602declare id_trans[trans]
3603
3604method "ηC" for Π :: ‹<'a::{AOT_Term_id_2,AOT_κs}>› = (match conclusion in "[v ⊨ τ{Π} = τ'{Π}]" for v τ τ' ⇒ ‹
3605rule "rule=E"[rotated 1, OF "eta-conversion-lemma1:2"[THEN "→E", of v "«[Π]»", symmetric]]
3606›)
3607(*
3608AOT_theorem ‹[λy [λz [P]z]y → [λu [S]u]y] = [λy [P]y → [S]y]›
3609  apply ("ηC" "«[P]»") defer
3610   apply ("ηC" "«[S]»") defer
3611  oops
3612*)
3613(* TODO: proper representation of eta_conversion_lemma2 *)
3614
3615AOT_theorem "sub-des-lam:1": ‹[λz1...zn  χ{z1...zn, ιx φ{x}}]↓ & ιx φ{x} = ιx ψ{x} → [λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, ιx ψ{x}}]›
3616proof(rule "→I")
3617  AOT_assume A: ‹[λz1...zn  χ{z1...zn, ιx φ{x}}]↓ & ιx φ{x} = ιx ψ{x}›
3618  AOT_show ‹[λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, ιx ψ{x}}]›
3619    using "rule=E"[where φ="λ τ . «[λz1...zn χ{z1...zn, ιx φ{x}}] = [λz1...zn χ{z1...zn, τ}]»",
3620               OF "=I"(1)[OF A[THEN "&E"(1)]], OF A[THEN "&E"(2)]]
3621    by blast
3622qed
3623
3624AOT_theorem "sub-des-lam:2": ‹ιx φ{x} = ιx ψ{x} → χ{ιx φ{x}} = χ{ιx ψ{x}}› for χ :: ‹κ ⇒ 𝗈›
3625  using "rule=E"[where φ="λ τ . «χ{ιx φ{x}} = χ{τ}»", OF "=I"(1)[OF "log-prop-prop:2"]] "→I" by blast
3626
3627AOT_theorem "prop-equiv": ‹F = G ≡ ∀x (x[F] ≡ x[G])›
3628proof(rule "≡I"; rule "→I")
3629  AOT_assume ‹F = G›
3630  AOT_thus ‹∀x (x[F] ≡ x[G])›
3631    by (rule "rule=E"[rotated]) (fact "oth-class-taut:3:a"[THEN GEN])
3632next
3633  AOT_assume ‹∀x (x[F] ≡ x[G])›
3634  AOT_hence ‹x[F] ≡ x[G]› for x using "∀E" by blast
3635  AOT_hence ‹□(x[F] ≡ x[G])› for x using "en-eq:6[1]"[THEN "≡E"(1)] by blast
3636  AOT_hence ‹∀x □(x[F] ≡ x[G])› by (rule GEN)
3637  AOT_hence ‹□∀x (x[F] ≡ x[G])› using BF[THEN "→E"] by fast
3638  AOT_thus "F = G" using "p-identity-thm2:1"[THEN "≡E"(2)] by blast
3639qed
3640
3641AOT_theorem "relations:1":
3642  assumes ‹INSTANCE_OF_CQT_2(φ)›
3643  shows ‹∃F □∀x1...∀xn ([F]x1...xn ≡ φ{x1...xn})›
3644  apply (rule "∃I"(1)[where τ="«[λx1...xn φ{x1...xn}]»"])
3645  using "cqt:2[lambda]"[OF assms, axiom_inst] "beta-C-cor:2"[THEN "→E", THEN RN] by blast+
3646
3647AOT_theorem "relations:2":
3648  assumes ‹INSTANCE_OF_CQT_2(φ)›
3649  shows ‹∃F □∀x ([F]x ≡ φ{x})›
3650  using "relations:1" assms by blast
3651
3652AOT_theorem "block-paradox:1": ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3653proof(rule RAA(2))
3654  let ?φ="λ τ. «∃G (τ[G] & ¬[G]τ)»"
3655  AOT_assume A: ‹[λx «?φ x»]↓›
3656  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ F = [λx «?φ x»]))›
3657    using "A-objects"[axiom_inst] by fast
3658  then AOT_obtain a where ξ: ‹A!a & ∀F (a[F] ≡ F = [λx «?φ x»])›
3659    using "∃E"[rotated] by blast
3660  AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3661  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3662    AOT_assume B: ‹[λx «?φ x»]a›
3663    AOT_hence ‹∃G (a[G] & ¬[G]a)› using "β→C" A by blast
3664    then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3665    moreover AOT_have ‹P = [λx «?φ x»]›
3666      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3667    ultimately AOT_have ‹¬[λx «?φ x»]a›
3668      using "rule=E" "&E"(2) by fast
3669    AOT_thus ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B RAA by blast
3670  next
3671    AOT_assume B: ‹¬[λx «?φ x»]a›
3672    AOT_hence ‹¬∃G (a[G] & ¬[G]a)› using "β←C" "cqt:2[const_var]"[of a, axiom_inst] A by blast
3673    AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3674    AOT_have ‹∀G (a[G] → [G]a)›
3675      by (AOT_subst_old "λ Π . «a[Π] → [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3676         (auto simp: "oth-class-taut:1:a" C)
3677    AOT_hence ‹a[λx «?φ x»] → [λx «?φ x»]a› using "∀E" A by blast
3678    moreover AOT_have ‹a[λx «?φ x»]› using ξ[THEN "&E"(2), THEN "∀E"(1), OF A, THEN "≡E"(2)]
3679      using "=I"(1)[OF A] by blast
3680    ultimately AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B "→E" RAA by blast
3681  qed
3682qed(simp)
3683
3684AOT_theorem "block-paradox:2": ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3685proof(rule RAA(2))
3686  AOT_assume ‹∃F ∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
3687  then AOT_obtain F where F_prop: ‹∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))› using "∃E"[rotated] by blast
3688  AOT_have ‹∃x (A!x & ∀G (x[G] ≡ G = F))›
3689    using "A-objects"[axiom_inst] by fast
3690  then AOT_obtain a where ξ: ‹A!a & ∀G (a[G] ≡ G = F)›
3691    using "∃E"[rotated] by blast
3692  AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3693  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3694    AOT_assume B: ‹[F]a›
3695    AOT_hence ‹∃G (a[G] & ¬[G]a)› using F_prop[THEN "∀E"(2), THEN "≡E"(1)] by blast
3696    then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3697    moreover AOT_have ‹P = F›
3698      using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3699    ultimately AOT_have ‹¬[F]a›
3700      using "rule=E" "&E"(2) by fast
3701    AOT_thus ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B RAA by blast
3702  next
3703    AOT_assume B: ‹¬[F]a›
3704    AOT_hence ‹¬∃G (a[G] & ¬[G]a)›
3705      using "oth-class-taut:4:b"[THEN "≡E"(1), OF F_prop[THEN "∀E"(2)[of _ _ a]], THEN "≡E"(1)] by simp
3706    AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3707    AOT_have ‹∀G (a[G] → [G]a)›
3708      by (AOT_subst_old "λ Π . «a[Π] → [Π]a»" "λ Π . «¬(a[Π] & ¬[Π]a)»")
3709         (auto simp: "oth-class-taut:1:a" C)
3710    AOT_hence ‹a[F] → [F]a› using "∀E" by blast
3711    moreover AOT_have ‹a[F]› using ξ[THEN "&E"(2), THEN "∀E"(2), of F, THEN "≡E"(2)]
3712      using "=I"(2) by blast
3713    ultimately AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B "→E" RAA by blast
3714  qed
3715qed(simp)
3716
3717AOT_theorem "block-paradox:3": ‹¬∀y [λz z = y]↓›
3718proof(rule RAA(2))
3719  AOT_assume θ: ‹∀y [λz z = y]↓›
3720  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃y(F = [λz z = y] & ¬y[F])))›
3721    using "A-objects"[axiom_inst] by force
3722  then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y (F = [λz z = y] & ¬y[F]))›
3723    using "∃E"[rotated] by blast
3724  AOT_have ζ: ‹a[λz z = a] ≡ ∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3725    using θ[THEN "∀E"(2)] a_prop[THEN "&E"(2), THEN "∀E"(1)] by blast
3726  AOT_show ‹¬∀y [λz z = y]↓›
3727  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3728    AOT_assume A: ‹a[λz z = a]›
3729    AOT_hence ‹∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3730      using ζ[THEN "≡E"(1)] by blast
3731    then AOT_obtain b where b_prop: ‹[λz z = a] = [λz z = b] & ¬b[λz z = a]›
3732      using "∃E"[rotated] by blast
3733    moreover AOT_have ‹a = a› by (rule "=I")
3734    moreover AOT_have ‹[λz z = a]↓› using θ "∀E" by blast
3735    moreover AOT_have ‹a↓› using "cqt:2[const_var]"[axiom_inst] .
3736    ultimately AOT_have ‹[λz z = a]a› using "β←C" by blast
3737    AOT_hence ‹[λz z = b]a› using "rule=E" b_prop[THEN "&E"(1)] by fast
3738    AOT_hence ‹a = b› using "β→C" by blast
3739    AOT_hence ‹b[λz z = a]› using A "rule=E" by fast
3740    AOT_thus ‹¬∀y [λz z = y]↓› using b_prop[THEN "&E"(2)] RAA by blast
3741  next
3742    AOT_assume A: ‹¬a[λz z = a]›
3743    AOT_hence ‹¬∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3744      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3745    AOT_hence ‹∀y ¬([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3746      using "cqt-further:4"[THEN "→E"] by blast
3747    AOT_hence ‹¬([λz z = a] = [λz z = a] & ¬a[λz z = a])›
3748      using "∀E" by blast
3749    AOT_hence ‹[λz z = a] = [λz z = a] → a[λz z = a]›
3750      by (metis "&I" "deduction-theorem" "raa-cor:4")
3751    AOT_hence ‹a[λz z = a]› using "=I"(1) θ[THEN "∀E"(2)] "→E" by blast
3752    AOT_thus ‹¬∀y [λz z = y]↓› using A RAA by blast
3753  qed
3754qed(simp)
3755
3756AOT_theorem "block-paradox:4": ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3757proof(rule RAA(2))
3758  AOT_assume θ: ‹∀y ∃F ∀x([F]x ≡ x = y)›
3759  AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F])))›
3760    using "A-objects"[axiom_inst] by force
3761  then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F]))›
3762    using "∃E"[rotated] by blast
3763  AOT_obtain F where F_prop: ‹∀x ([F]x ≡ x = a)› using θ[THEN "∀E"(2)] "∃E"[rotated] by blast
3764  AOT_have ζ: ‹a[F] ≡ ∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3765    using a_prop[THEN "&E"(2), THEN "∀E"(2)] by blast
3766  AOT_show ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3767  proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3768    AOT_assume A: ‹a[F]›
3769    AOT_hence ‹∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3770      using ζ[THEN "≡E"(1)] by blast
3771    then AOT_obtain b where b_prop: ‹∀y ([F]y ≡ y = b) & ¬b[F]›
3772      using "∃E"[rotated] by blast
3773    moreover AOT_have ‹[F]a› using F_prop[THEN "∀E"(2), THEN "≡E"(2)] "=I"(2) by blast
3774    ultimately AOT_have ‹a = b› using "∀E"(2) "≡E"(1) "&E" by fast
3775    AOT_hence ‹a = b› using "β→C" by blast
3776    AOT_hence ‹b[F]› using A "rule=E" by fast
3777    AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using b_prop[THEN "&E"(2)] RAA by blast
3778  next
3779    AOT_assume A: ‹¬a[F]›
3780    AOT_hence ‹¬∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3781      using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3782    AOT_hence ‹∀z ¬(∀y ([F]y ≡ y = z) & ¬z[F])›
3783      using "cqt-further:4"[THEN "→E"] by blast
3784    AOT_hence ‹¬(∀y ([F]y ≡ y = a) & ¬a[F])›
3785      using "∀E" by blast
3786    AOT_hence ‹∀y ([F]y ≡ y = a) → a[F]›
3787      by (metis "&I" "deduction-theorem" "raa-cor:4")
3788    AOT_hence ‹a[F]› using F_prop "→E" by blast
3789    AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using A RAA by blast
3790  qed
3791qed(simp)
3792
3793AOT_theorem "block-paradox:5": ‹¬∃F∀x∀y([F]xy ≡ y = x)›
3794proof(rule "raa-cor:2")
3795  AOT_assume ‹∃F∀x∀y([F]xy ≡ y = x)›
3796  then AOT_obtain F where F_prop: ‹∀x∀y([F]xy ≡ y = x)› using "∃E"[rotated] by blast
3797  {
3798    fix x
3799    AOT_have 1: ‹∀y([F]xy ≡ y = x)› using F_prop "∀E" by blast
3800    AOT_have 2: ‹[λz [F]xz]↓› by "cqt:2[lambda]"
3801    moreover AOT_have ‹∀y([λz [F]xz]y ≡ y = x)›
3802    proof(rule "∀I")
3803      fix y
3804      AOT_have ‹[λz [F]xz]y ≡ [F]xy›
3805        using "beta-C-meta"[THEN "→E"] 2 by fast
3806      also AOT_have ‹... ≡ y = x› using 1 "∀E"
3807        by fast
3808      finally AOT_show ‹[λz [F]xz]y ≡ y = x›.
3809    qed
3810    ultimately AOT_have ‹∃F∀y([F]y ≡ y = x)›
3811      using "∃I" by fast
3812  }
3813  AOT_hence ‹∀x∃F∀y([F]y ≡ y = x)›
3814    by (rule GEN)
3815  AOT_thus ‹∀x∃F∀y([F]y ≡ y = x) & ¬∀x∃F∀y([F]y ≡ y = x)›
3816    using "&I" "block-paradox:4" by blast
3817qed
3818
3819AOT_act_theorem "block-paradox2:1": ‹∀x [G]x → ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3820proof(rule "→I"; rule "raa-cor:2")
3821  AOT_assume antecedant: ‹∀x [G]x›
3822  AOT_have Lemma: ‹∀x ([G]ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
3823  proof(rule GEN)
3824    fix x
3825    AOT_have A: ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3826    proof(rule "≡I"; rule "→I")
3827      AOT_assume ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x))›
3828      AOT_hence ‹ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
3829        using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
3830      AOT_thus ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3831        using "1-exists:1"[THEN "≡E"(1)] by blast
3832    next
3833      AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3834      AOT_obtain a where a_1: ‹a = x & ∃H (x[H] & ¬[H]x)› and a_2: ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = a)›
3835        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3836      AOT_have a_3: ‹[G]a›
3837        using antecedant "∀E" by blast
3838      AOT_show ‹[G]ιy (y = x & ∃H (x[H] & ¬[H]x))›
3839        apply (rule "russell-axiom[exe,1].russell-axiom"[THEN "≡E"(2)])
3840        apply (rule "∃I"(2))
3841        using a_1 a_2 a_3 "&I" by blast
3842    qed
3843    also AOT_have B: ‹... ≡ ∃H (x[H] & ¬[H]x)›
3844    proof (rule "≡I"; rule "→I")
3845      AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3846      AOT_obtain a where ‹a = x & ∃H (x[H] & ¬[H]x)›
3847        using "uniqueness:1"[THEN "≡dfE", OF A] "&E" "∃E"[rotated] by blast
3848      AOT_thus ‹∃H (x[H] & ¬[H]x)› using "&E" by blast
3849    next
3850      AOT_assume ‹∃H (x[H] & ¬[H]x)›
3851      AOT_hence ‹x = x & ∃H (x[H] & ¬[H]x)›
3852        using "id-eq:1" "&I" by blast
3853      moreover AOT_have ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = x)›
3854        by (simp add: "Conjunction Simplification"(1) "universal-cor")
3855      ultimately AOT_show ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3856        using "uniqueness:1"[THEN "≡dfI"] "&I" "∃I"(2) by fast
3857    qed
3858    finally AOT_show ‹([G]ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))› .
3859  qed
3860
3861  AOT_assume A: ‹[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3862  AOT_have θ: ‹∀x ([λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ [G]ιy(y = x & ∃H (x[H] & ¬[H]x)))›
3863    using "beta-C-meta"[THEN "→E", OF A] "∀I" by fast
3864  AOT_have ‹∀x ([λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ ∃H (x[H] & ¬[H]x))›
3865    using θ Lemma "cqt-basic:10"[THEN "→E"] "&I" by fast
3866  AOT_hence ‹∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))›
3867    using "∃I"(1) A by fast
3868  AOT_thus ‹(∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))) & (¬∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x)))›
3869    using "block-paradox:2" "&I" by blast
3870qed
3871
3872AOT_act_theorem "block-paradox2:2": ‹∃G ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3873proof(rule "∃I"(1))
3874  AOT_have 0: ‹[λx ∀p (p →p)]↓›
3875    by "cqt:2[lambda]"
3876  moreover AOT_have ‹∀x [λx ∀p (p →p)]x›
3877    apply (rule GEN)
3878    apply (rule "beta-C-cor:2"[THEN "→E", OF 0, THEN "∀E"(2), THEN "≡E"(2)])
3879    using "if-p-then-p" GEN by fast
3880  moreover AOT_have ‹∀G (∀x [G]x → ¬[λx [G]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓)›
3881      using "block-paradox2:1" "∀I" by fast
3882  ultimately AOT_show ‹¬[λx [λx ∀p (p →p)]ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3883    using "∀E"(1) "→E" by blast
3884qed("cqt:2[lambda]")
3885
3886AOT_theorem propositions: ‹∃p □(p ≡ φ)›
3887proof(rule "∃I"(1))
3888  AOT_show ‹□(φ ≡ φ)›
3889    by (simp add: RN "oth-class-taut:3:a")
3890next
3891  AOT_show ‹φ↓›
3892    by (simp add: "log-prop-prop:2")
3893qed
3894
3895AOT_theorem "pos-not-equiv-ne:1": ‹(◇¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)) → F ≠ G›
3896proof (rule "→I")
3897  AOT_assume ‹◇¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)›
3898  AOT_hence ‹¬□∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)›
3899    using "KBasic:11"[THEN "≡E"(2)] by blast
3900  AOT_hence ‹¬(F = G)›
3901    using "id-rel-nec-equiv:1" "modus-tollens:1" by blast
3902  AOT_thus ‹F ≠ G›
3903    using "=-infix"[THEN "≡dfI"] by blast
3904qed
3905
3906AOT_theorem "pos-not-equiv-ne:2": ‹(◇¬(φ{F} ≡ φ{G})) → F ≠ G›
3907proof (rule "→I")
3908  AOT_modally_strict {
3909    AOT_have ‹¬(φ{F} ≡ φ{G}) → ¬(F = G)›
3910    proof (rule "→I"; rule "raa-cor:2")
3911      AOT_assume 1: ‹F = G›
3912      AOT_hence ‹φ{F} → φ{G}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3913      moreover {
3914        AOT_have ‹G = F› using 1 id_sym by blast
3915        AOT_hence ‹φ{G} → φ{F}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3916      }
3917      ultimately AOT_have ‹φ{F} ≡ φ{G}› using "≡I" by blast
3918      moreover AOT_assume ‹¬(φ{F} ≡ φ{G})›
3919      ultimately AOT_show ‹(φ{F} ≡ φ{G}) & ¬(φ{F} ≡ φ{G})›
3920        using "&I" by blast
3921    qed
3922  }
3923  AOT_hence ‹◇¬(φ{F} ≡ φ{G}) → ◇¬(F = G)›
3924    using "RM:2[prem]" by blast
3925  moreover AOT_assume ‹◇¬(φ{F} ≡ φ{G})›
3926  ultimately AOT_have 0: ‹◇¬(F = G)› using "→E" by blast
3927  AOT_have ‹◇(F ≠ G)›
3928    by (AOT_subst ‹F ≠ G› ‹¬(F = G)›)
3929       (auto simp: "=-infix" "≡Df" 0)
3930  AOT_thus ‹F ≠ G›
3931    using "id-nec2:3"[THEN "→E"] by blast
3932qed
3933
3934AOT_theorem "pos-not-equiv-ne:2[zero]": ‹(◇¬(φ{p} ≡ φ{q})) → p ≠ q›
3935proof (rule "→I")
3936  AOT_modally_strict {
3937    AOT_have ‹¬(φ{p} ≡ φ{q}) → ¬(p = q)›
3938    proof (rule "→I"; rule "raa-cor:2")
3939      AOT_assume 1: ‹p = q›
3940      AOT_hence ‹φ{p} → φ{q}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3941      moreover {
3942        AOT_have ‹q = p› using 1 id_sym by blast
3943        AOT_hence ‹φ{q} → φ{p}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3944      }
3945      ultimately AOT_have ‹φ{p} ≡ φ{q}› using "≡I" by blast
3946      moreover AOT_assume ‹¬(φ{p} ≡ φ{q})›
3947      ultimately AOT_show ‹(φ{p} ≡ φ{q}) & ¬(φ{p} ≡ φ{q})›
3948        using "&I" by blast
3949    qed
3950  }
3951  AOT_hence ‹◇¬(φ{p} ≡ φ{q}) → ◇¬(p = q)›
3952    using "RM:2[prem]" by blast
3953  moreover AOT_assume ‹◇¬(φ{p} ≡ φ{q})›
3954  ultimately AOT_have 0: ‹◇¬(p = q)› using "→E" by blast
3955  AOT_have ‹◇(p ≠ q)›
3956    by (AOT_subst ‹p ≠ q› ‹¬(p = q)›)
3957       (auto simp: 0 "=-infix" "≡Df")
3958  AOT_thus ‹p ≠ q›
3959    using "id-nec2:3"[THEN "→E"] by blast
3960qed
3961
3962AOT_theorem "pos-not-equiv-ne:3": ‹(¬∀x1...∀xn ([F]x1...xn ≡ [G]x1...xn)) → F ≠ G›
3963  using "→I" "pos-not-equiv-ne:1"[THEN "→E"] "T◇"[THEN "→E"] by blast
3964
3965AOT_theorem "pos-not-equiv-ne:4": ‹(¬(φ{F} ≡ φ{G})) → F ≠ G›
3966  using "→I" "pos-not-equiv-ne:2"[THEN "→E"] "T◇"[THEN "→E"] by blast
3967
3968AOT_theorem "pos-not-equiv-ne:4[zero]": ‹(¬(φ{p} ≡ φ{q})) → p ≠ q›
3969  using "→I" "pos-not-equiv-ne:2[zero]"[THEN "→E"] "T◇"[THEN "→E"] by blast
3970
3971AOT_define relation_negation :: "Π ⇒ Π" ("_-")
3972  "df-relation-negation": "[F]- =df [λx1...xn ¬[F]x1...xn]"
3973
3974nonterminal φneg
3975syntax "" :: "φneg ⇒ τ" ("_")
3976syntax "" :: "φneg ⇒ φ" ("'(_')")
3977
3978AOT_define relation_negation_0 :: ‹φ ⇒ φneg› ("'(_')-")
3979  "df-relation-negation[zero]": "(p)- =df [λ ¬p]"
3980
3981AOT_theorem "rel-neg-T:1": ‹[λx1...xn ¬[Π]x1...xn]↓›
3982  by "cqt:2[lambda]"
3983
3984AOT_theorem "rel-neg-T:1[zero]": ‹[λ ¬φ]↓›
3985  using "cqt:2[lambda0]"[axiom_inst] by blast
3986
3987AOT_theorem "rel-neg-T:2": ‹[Π]- = [λx1...xn ¬[Π]x1...xn]›
3988  using "=I"(1)[OF "rel-neg-T:1"]
3989  by (rule "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"])
3990
3991AOT_theorem "rel-neg-T:2[zero]": ‹(φ)- = [λ ¬φ]›
3992  using "=I"(1)[OF "rel-neg-T:1[zero]"]
3993  by (rule "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"])
3994
3995AOT_theorem "rel-neg-T:3": ‹[Π]-↓›
3996  using "=dfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"] "rel-neg-T:1" by blast
3997
3998AOT_theorem "rel-neg-T:3[zero]": ‹(φ)-↓›
3999  using "log-prop-prop:2" by blast
4000(*  using "=dfI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"] "rel-neg-T:1[zero]" by blast *)
4001
4002(* Note: PLM states the zero place case twice *)
4003AOT_theorem "thm-relation-negation:1": ‹[F]-x1...xn ≡ ¬[F]x1...xn›
4004proof -
4005  AOT_have ‹[F]-x1...xn ≡ [λx1...xn ¬[F]x1...xn]x1...xn›
4006    using "rule=E"[rotated, OF "rel-neg-T:2"] "rule=E"[rotated, OF "rel-neg-T:2"[THEN id_sym]]
4007    "→I" "≡I" by fast
4008  also AOT_have ‹... ≡ ¬[F]x1...xn›
4009    using "beta-C-meta"[THEN "→E", OF "rel-neg-T:1"] by fast
4010  finally show ?thesis.
4011qed
4012
4013AOT_theorem "thm-relation-negation:2": ‹¬[F]-x1...xn ≡ [F]x1...xn›
4014  apply (AOT_subst ‹[F]x1...xn› ‹¬¬[F]x1...xn›)
4015   apply (simp add: "oth-class-taut:3:b")
4016  apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4017  using "thm-relation-negation:1".
4018
4019AOT_theorem "thm-relation-negation:3": ‹((p)-) ≡ ¬p›
4020proof -
4021  AOT_have ‹(p)- = [λ ¬p]› using "rel-neg-T:2[zero]" by blast
4022  AOT_hence ‹((p)-) ≡ [λ ¬p]›
4023    using "df-relation-negation[zero]" "log-prop-prop:2" "oth-class-taut:3:a" "rule-id-df:2:a" by blast
4024  also AOT_have ‹[λ ¬p] ≡ ¬p›
4025    by (simp add: "propositions-lemma:2")
4026  finally show ?thesis.
4027qed
4028
4029AOT_theorem "thm-relation-negation:4": ‹(¬((p)-)) ≡ p›
4030  using "thm-relation-negation:3"[THEN "≡E"(1)]
4031        "thm-relation-negation:3"[THEN "≡E"(2)]
4032        "≡I" "→I" RAA by metis
4033
4034AOT_theorem "thm-relation-negation:5": ‹[F] ≠ [F]-›
4035proof -
4036  AOT_have ‹¬([F] = [F]-)›
4037  proof (rule RAA(2))
4038    AOT_show ‹[F]x1...xn → [F]x1...xn› for x1xn
4039      using "if-p-then-p".
4040  next
4041    AOT_assume ‹[F] = [F]-›
4042    AOT_hence ‹[F]- = [F]› using id_sym by blast
4043    AOT_hence ‹[F]x1...xn ≡ ¬[F]x1...xn› for x1xn
4044      using "rule=E" "thm-relation-negation:1" by fast
4045    AOT_thus ‹¬([F]x1...xn → [F]x1...xn)› for x1xn
4046      using "≡E" RAA by metis
4047  qed
4048  thus ?thesis
4049    using "≡dfI" "=-infix" by blast
4050qed
4051
4052AOT_theorem "thm-relation-negation:6": ‹p ≠ (p)-›
4053proof -
4054  AOT_have ‹¬(p = (p)-)›
4055  proof (rule RAA(2))
4056    AOT_show ‹p → p›
4057      using "if-p-then-p".
4058  next
4059    AOT_assume ‹p = (p)-›
4060    AOT_hence ‹(p)- = p› using id_sym by blast
4061    AOT_hence ‹p ≡ ¬p›
4062      using "rule=E" "thm-relation-negation:3" by fast
4063    AOT_thus ‹¬(p → p)›
4064      using "≡E" RAA by metis
4065  qed
4066  thus ?thesis
4067    using "≡dfI" "=-infix" by blast
4068qed
4069
4070AOT_theorem "thm-relation-negation:7": ‹(p)- = (¬p)›
4071  apply (rule "df-relation-negation[zero]"[THEN "=dfE"(1)])
4072  using "cqt:2[lambda0]"[axiom_inst] "rel-neg-T:2[zero]" "propositions-lemma:1" id_trans by blast+
4073
4074AOT_theorem "thm-relation-negation:8": ‹p = q → (¬p) = (¬q)›
4075proof(rule "→I")
4076  AOT_assume ‹p = q›
4077  moreover AOT_have ‹(¬p)↓› using "log-prop-prop:2".
4078  moreover AOT_have ‹(¬p) = (¬p)› using calculation(2) "=I" by blast
4079  ultimately AOT_show ‹(¬p) = (¬q)›
4080    using "rule=E" by fast
4081qed
4082
4083AOT_theorem "thm-relation-negation:9": ‹p = q → (p)- = (q)-›
4084proof(rule "→I")
4085  AOT_assume ‹p = q›
4086  AOT_hence ‹(¬p) = (¬q)› using "thm-relation-negation:8" "→E" by blast
4087  AOT_thus ‹(p)- = (q)-›
4088    using "thm-relation-negation:7" id_sym id_trans by metis
4089qed
4090
4091AOT_define Necessary :: ‹Π ⇒ φ› ("Necessary'(_')")
4092  "contingent-properties:1": ‹Necessary([F]) ≡df □∀x1...∀xn [F]x1...xn›
4093
4094AOT_define Necessary0 :: ‹φ ⇒ φ› ("Necessary0'(_')")
4095  "contingent-properties:1[zero]": ‹Necessary0(p) ≡df □p›
4096
4097AOT_define Impossible :: ‹Π ⇒ φ› ("Impossible'(_')")
4098  "contingent-properties:2": ‹Impossible([F]) ≡df F↓ & □∀x1...∀xn ¬[F]x1...xn›
4099
4100AOT_define Impossible0 :: ‹φ ⇒ φ› ("Impossible0'(_')")
4101  "contingent-properties:2[zero]": ‹Impossible0(p) ≡df □¬p›
4102
4103AOT_define NonContingent :: ‹Π ⇒ φ› ("NonContingent'(_')")
4104  "contingent-properties:3": ‹NonContingent([F]) ≡df Necessary([F]) ∨ Impossible([F])›
4105
4106AOT_define NonContingent0 :: ‹φ ⇒ φ› ("NonContingent0'(_')")
4107  "contingent-properties:3[zero]": ‹NonContingent0(p) ≡df Necessary0(p) ∨ Impossible0(p)›
4108
4109AOT_define Contingent :: ‹Π ⇒ φ› ("Contingent'(_')")
4110  "contingent-properties:4": ‹Contingent([F]) ≡df F↓ & ¬(Necessary([F]) ∨ Impossible([F]))›
4111
4112AOT_define Contingent0 :: ‹φ ⇒ φ› ("Contingent0'(_')")
4113  "contingent-properties:4[zero]": ‹Contingent0(p) ≡df ¬(Necessary0(p) ∨ Impossible0(p))›
4114
4115
4116AOT_theorem "thm-cont-prop:1": ‹NonContingent([F]) ≡ NonContingent([F]-)›
4117proof (rule "≡I"; rule "→I")
4118  AOT_assume ‹NonContingent([F])›
4119  AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4120    using "≡dfE"[OF "contingent-properties:3"] by blast
4121  moreover {
4122    AOT_assume ‹Necessary([F])›
4123    AOT_hence ‹□(∀x1...∀xn [F]x1...xn)›
4124      using "≡dfE"[OF "contingent-properties:1"] by blast
4125    moreover AOT_modally_strict {
4126      AOT_assume ‹∀x1...∀xn [F]x1...xn›
4127      AOT_hence ‹[F]x1...xn› for x1xn using "∀E" by blast
4128      AOT_hence ‹¬[F]-x1...xn› for x1xn
4129        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:2" "≡E"(1))
4130      AOT_hence ‹∀x1...∀xn ¬[F]-x1...xn› using "∀I" by fast
4131    }
4132    ultimately AOT_have ‹□(∀x1...∀xn ¬[F]-x1...xn)›
4133      using "RN[prem]"[where Γ="{«∀x1...∀xn [F]x1...xn»}", simplified] by blast
4134    AOT_hence ‹Impossible([F]-)›
4135      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(2)]
4136      by blast
4137  }
4138  moreover {
4139    AOT_assume ‹Impossible([F])›
4140    AOT_hence ‹□(∀x1...∀xn ¬[F]x1...xn)›
4141      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)]
4142      by blast
4143    moreover AOT_modally_strict {
4144      AOT_assume ‹∀x1...∀xn ¬[F]x1...xn›
4145      AOT_hence ‹¬[F]x1...xn› for x1xn using "∀E" by blast
4146      AOT_hence ‹[F]-x1...xn› for x1xn
4147        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(1))
4148      AOT_hence ‹∀x1...∀xn [F]-x1...xn› using "∀I" by fast
4149    }
4150    ultimately AOT_have ‹□(∀x1...∀xn [F]-x1...xn)›
4151      using "RN[prem]"[where Γ="{«∀x1...∀xn ¬[F]x1...xn»}"] by blast
4152    AOT_hence ‹Necessary([F]-)›
4153      using "≡dfI"[OF "contingent-properties:1"] by blast
4154  }
4155  ultimately AOT_have ‹Necessary([F]-) ∨ Impossible([F]-)›
4156    using "∨E"(1) "∨I" "→I" by metis
4157  AOT_thus ‹NonContingent([F]-)›
4158    using "≡dfI"[OF "contingent-properties:3"] by blast
4159next
4160  AOT_assume ‹NonContingent([F]-)›
4161  AOT_hence ‹Necessary([F]-) ∨ Impossible([F]-)›
4162    using "≡dfE"[OF "contingent-properties:3"] by blast
4163  moreover {
4164    AOT_assume ‹Necessary([F]-)›
4165    AOT_hence ‹□(∀x1...∀xn [F]-x1...xn)›
4166      using "≡dfE"[OF "contingent-properties:1"] by blast
4167    moreover AOT_modally_strict {
4168      AOT_assume ‹∀x1...∀xn [F]-x1...xn›
4169      AOT_hence ‹[F]-x1...xn› for x1xn using "∀E" by blast
4170      AOT_hence ‹¬[F]x1...xn› for x1xn
4171        by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(2))
4172      AOT_hence ‹∀x1...∀xn ¬[F]x1...xn› using "∀I" by fast
4173    }
4174    ultimately AOT_have ‹□∀x1...∀xn ¬[F]x1...xn›
4175      using "RN[prem]"[where Γ="{«∀x1...∀xn [F]-x1...xn»}"] by blast
4176    AOT_hence ‹Impossible([F])›
4177      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(2)]
4178      by blast
4179  }
4180  moreover {
4181    AOT_assume ‹Impossible([F]-)›
4182    AOT_hence ‹□(∀x1...∀xn ¬[F]-x1...xn)›
4183      using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(1)]
4184      by blast
4185    moreover AOT_modally_strict {
4186      AOT_assume ‹∀x1...∀xn ¬[F]-x1...xn›
4187      AOT_hence ‹¬[F]-x1...xn› for x1xn using "∀E" by blast
4188      AOT_hence ‹[F]x1...xn› for x1xn 
4189        using "thm-relation-negation:1"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4190              "useful-tautologies:1"[THEN "→E"] by blast
4191      AOT_hence ‹∀x1...∀xn [F]x1...xn› using "∀I" by fast
4192    }
4193    ultimately AOT_have ‹□(∀x1...∀xn [F]x1...xn)›
4194      using "RN[prem]"[where Γ="{«∀x1...∀xn ¬[F]-x1...xn»}"] by blast
4195    AOT_hence ‹Necessary([F])›
4196      using "≡dfI"[OF "contingent-properties:1"] by blast
4197  }
4198  ultimately AOT_have ‹Necessary([F]) ∨ Impossible([F])›
4199    using "∨E"(1) "∨I" "→I" by metis
4200  AOT_thus ‹NonContingent([F])›
4201    using "≡dfI"[OF "contingent-properties:3"] by blast
4202qed
4203
4204AOT_theorem "thm-cont-prop:2": ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4205proof -
4206  AOT_have ‹Contingent([F]) ≡ ¬(Necessary([F]) ∨ Impossible([F]))›
4207    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]]
4208    by blast
4209  also AOT_have ‹... ≡ ¬Necessary([F]) & ¬Impossible([F])›
4210    using "oth-class-taut:5:d" by fastforce
4211  also AOT_have ‹... ≡ ¬Impossible([F]) & ¬Necessary([F])›
4212    by (simp add: "Commutativity of &")
4213  also AOT_have ‹... ≡ ◇∃x [F]x & ¬Necessary([F])›
4214  proof (rule "oth-class-taut:4:e"[THEN "→E"])
4215    AOT_have ‹¬Impossible([F]) ≡ ¬□¬ ∃x [F]x›
4216      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4217      apply (AOT_subst ‹∃x [F]x› ‹¬ ∀x ¬[F]x›)
4218       apply (simp add: "conventions:4" "≡Df")
4219      apply (AOT_subst (reverse) ‹¬¬∀x ¬[F]x› ‹∀x ¬[F]x›)
4220       apply (simp add: "oth-class-taut:3:b")
4221      using "contingent-properties:2"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]] by blast
4222    also AOT_have ‹... ≡ ◇∃x [F]x›
4223      using "conventions:5"[THEN "≡Df", symmetric] by blast
4224    finally AOT_show ‹¬Impossible([F]) ≡ ◇∃x [F]x› .
4225  qed
4226  also AOT_have ‹... ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4227  proof (rule "oth-class-taut:4:f"[THEN "→E"])
4228    AOT_have ‹¬Necessary([F]) ≡ ¬□¬∃x ¬[F]x›
4229      apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4230      apply (AOT_subst ‹∃x ¬[F]x› ‹¬ ∀x ¬¬[F]x›)
4231       apply (simp add: "conventions:4" "≡Df")
4232      apply (AOT_subst (reverse) ‹¬¬[F]x› ‹[F]x› bound: x)
4233       apply (simp add: "oth-class-taut:3:b")
4234      apply (AOT_subst (reverse) ‹¬¬∀x [F]x› ‹∀x [F]x›)
4235      by (auto simp: "oth-class-taut:3:b" "contingent-properties:1" "≡Df")
4236    also AOT_have ‹... ≡ ◇∃x ¬[F]x›
4237      using "conventions:5"[THEN "≡Df", symmetric] by blast
4238    finally AOT_show ‹¬Necessary([F]) ≡ ◇∃x ¬[F]x›.
4239  qed
4240  finally show ?thesis.
4241qed
4242
4243AOT_theorem "thm-cont-prop:3": ‹Contingent([F]) ≡ Contingent([F]-)› for F::‹<κ> AOT_var›
4244proof -
4245  {
4246    fix Π :: ‹<κ>›
4247    AOT_assume ‹Π↓›
4248    moreover AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4249      using "thm-cont-prop:2" GEN by fast
4250    ultimately  AOT_have ‹Contingent([Π]) ≡ ◇∃x [Π]x & ◇∃x ¬[Π]x›
4251      using "thm-cont-prop:2" "∀E" by fast
4252  } note 1 = this
4253  AOT_have ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4254    using "thm-cont-prop:2" by blast
4255  also AOT_have ‹... ≡ ◇∃x ¬[F]x & ◇∃x [F]x›
4256    by (simp add: "Commutativity of &")
4257  also AOT_have ‹... ≡ ◇∃x [F]-x & ◇∃x [F]x›
4258    by (AOT_subst ‹[F]-x› ‹¬[F]x› bound: x)
4259       (auto simp: "thm-relation-negation:1" "oth-class-taut:3:a")
4260  also AOT_have ‹... ≡ ◇∃x [F]-x & ◇∃x ¬[F]-x›
4261    by (AOT_subst (reverse) ‹[F]x› ‹¬[F]-x› bound: x)
4262       (auto simp: "thm-relation-negation:2" "oth-class-taut:3:a")
4263  also AOT_have ‹... ≡ Contingent([F]-)›
4264    using 1[OF "rel-neg-T:3", symmetric] by blast
4265  finally show ?thesis.
4266qed
4267
4268AOT_define concrete_if_concrete :: ‹Π› ("L")  L_def: ‹L =df [λx E!x → E!x]›
4269
4270AOT_theorem "thm-noncont-e-e:1": ‹Necessary(L)›
4271proof -
4272  AOT_modally_strict {
4273    fix x
4274    AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4275    moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4276    moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4277    ultimately AOT_have ‹[λx E!x → E!x]x›
4278      using "β←C" by blast
4279  }
4280  AOT_hence 0: ‹□∀x [λx E!x → E!x]x›
4281    using RN GEN by blast
4282  show ?thesis
4283    apply (rule "=dfI"(2)[OF L_def])
4284     apply "cqt:2[lambda]"
4285    by (rule "contingent-properties:1"[THEN "≡dfI", OF 0])
4286qed
4287
4288AOT_theorem "thm-noncont-e-e:2": ‹Impossible([L]-)›
4289proof -
4290  AOT_modally_strict {
4291    fix x
4292
4293    AOT_have 0: ‹∀F (¬[F]-x ≡ [F]x)›
4294      using "thm-relation-negation:2" GEN by fast
4295    AOT_have ‹¬[λx E!x → E!x]-x ≡ [λx E!x → E!x]x›
4296      by (rule 0[THEN "∀E"(1)]) "cqt:2[lambda]"
4297    moreover {
4298      AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4299      moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4300      moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4301      ultimately AOT_have ‹[λx E!x → E!x]x›
4302        using "β←C" by blast
4303    }
4304    ultimately AOT_have ‹¬[λx E!x → E!x]-x›
4305      using "≡E" by blast
4306  }
4307  AOT_hence 0: ‹□∀x ¬[λx E!x → E!x]-x›
4308    using RN GEN by fast
4309  show ?thesis
4310    apply (rule "=dfI"(2)[OF L_def])
4311     apply "cqt:2[lambda]"
4312    apply (rule "contingent-properties:2"[THEN "≡dfI"]; rule "&I")
4313     using "rel-neg-T:3"
4314     apply blast
4315    using 0
4316    by blast
4317qed
4318
4319AOT_theorem "thm-noncont-e-e:3": ‹NonContingent(L)›
4320  using "thm-noncont-e-e:1"
4321  by (rule "contingent-properties:3"[THEN "≡dfI", OF "∨I"(1)])
4322
4323AOT_theorem "thm-noncont-e-e:4": ‹NonContingent([L]-)›
4324proof -
4325  AOT_have 0: ‹∀F (NonContingent([F]) ≡ NonContingent([F]-))›
4326    using "thm-cont-prop:1" "∀I" by fast
4327  moreover AOT_have 1: ‹L↓›
4328    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4329  AOT_show ‹NonContingent([L]-)›
4330    using "∀E"(1)[OF 0, OF 1, THEN "≡E"(1), OF "thm-noncont-e-e:3"] by blast
4331qed
4332
4333AOT_theorem "thm-noncont-e-e:5": ‹∃F ∃G (F ≠ «G::<κ>» & NonContingent([F]) & NonContingent([G]))›
4334proof (rule "∃I")+
4335  {
4336    AOT_have ‹∀F [F] ≠ [F]-› using "thm-relation-negation:5" GEN by fast
4337    moreover AOT_have ‹L↓›
4338      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4339    ultimately AOT_have ‹L ≠ [L]-› using "∀E" by blast
4340  }
4341  AOT_thus ‹L ≠ [L]- & NonContingent(L) & NonContingent([L]-)›
4342    using "thm-noncont-e-e:3" "thm-noncont-e-e:4" "&I" by metis
4343next
4344  AOT_show ‹[L]-↓›
4345    using "rel-neg-T:3" by blast
4346next
4347  AOT_show ‹L↓›
4348      by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
4349qed
4350
4351AOT_theorem "lem-cont-e:1": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4352proof -
4353  AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ∃x ◇([F]x & ◇¬[F]x)›
4354    using "BF◇" "CBF◇" "≡I" by blast
4355  also AOT_have ‹… ≡ ∃x (◇[F]x &  ◇¬[F]x)›
4356    by (AOT_subst ‹◇([F]x & ◇¬[F]x)› ‹◇[F]x &  ◇¬[F]x› bound: x)
4357       (auto simp: "S5Basic:11" "cqt-further:7")
4358  also AOT_have ‹… ≡ ∃x (◇¬[F]x & ◇[F]x)›
4359    by (AOT_subst ‹◇¬[F]x & ◇[F]x›  ‹◇[F]x & ◇¬[F]x› bound: x)
4360       (auto simp: "Commutativity of &" "cqt-further:7")
4361  also AOT_have ‹… ≡ ∃x ◇(¬[F]x & ◇[F]x)›
4362    by (AOT_subst ‹◇(¬[F]x & ◇[F]x)› ‹◇¬[F]x & ◇[F]x› bound: x)
4363       (auto simp: "S5Basic:11" "oth-class-taut:3:a")
4364  also AOT_have ‹… ≡ ◇∃x (¬[F]x & ◇[F]x)›
4365    using "BF◇" "CBF◇" "≡I" by fast
4366  finally show ?thesis.
4367qed
4368
4369AOT_theorem "lem-cont-e:2": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x ([F]-x & ◇¬[F]-x)›
4370proof -
4371  AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4372    using "lem-cont-e:1".
4373  also AOT_have ‹… ≡ ◇∃x ([F]-x & ◇¬[F]-x)›
4374    apply (AOT_subst ‹¬[F]-x› ‹[F]x› bound: x)
4375     apply (simp add: "thm-relation-negation:2")
4376    apply (AOT_subst ‹[F]-x› ‹¬[F]x› bound: x)
4377     apply (simp add: "thm-relation-negation:1")
4378    by (simp add: "oth-class-taut:3:a")
4379  finally show ?thesis.
4380qed
4381
4382AOT_theorem "thm-cont-e:1": ‹◇∃x (E!x & ◇¬E!x)›
4383proof (rule "CBF◇"[THEN "→E"])
4384  AOT_have ‹∃x ◇(E!x & ¬𝒜E!x)› using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] by blast
4385  then AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
4386  AOT_hence θ: ‹◇E!a & ◇¬𝒜E!a›
4387    using "KBasic2:3"[THEN "→E"] by blast
4388  AOT_have ξ: ‹◇E!a & ◇𝒜¬E!a›
4389    by (AOT_subst  ‹𝒜¬E!a› ‹¬𝒜E!a›)
4390       (auto simp: "logic-actual-nec:1"[axiom_inst] θ)
4391  AOT_have ζ: ‹◇E!a & 𝒜¬E!a›
4392    by (AOT_subst ‹𝒜¬E!a› ‹◇𝒜¬E!a›)
4393       (auto simp add: "Act-Sub:4" ξ)
4394  AOT_hence ‹◇E!a & ◇¬E!a›
4395    using "&E" "&I" "Act-Sub:3"[THEN "→E"] by blast
4396  AOT_hence ‹◇(E!a & ◇¬E!a)› using "S5Basic:11"[THEN "≡E"(2)] by simp
4397  AOT_thus ‹∃x ◇(E!x & ◇¬E!x)› using "∃I"(2) by fast
4398qed
4399
4400AOT_theorem "thm-cont-e:2": ‹◇∃x (¬E!x & ◇E!x)›
4401proof -
4402  AOT_have ‹∀F (◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x))›
4403    using "lem-cont-e:1" GEN by fast
4404  AOT_hence ‹(◇∃x (E!x & ◇¬E!x) ≡ ◇∃x (¬E!x & ◇E!x))›
4405    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4406  thus ?thesis using "thm-cont-e:1" "≡E" by blast
4407qed
4408
4409AOT_theorem "thm-cont-e:3": ‹◇∃x E!x›
4410proof (rule "CBF◇"[THEN "→E"])
4411  AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4412    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4413  AOT_hence ‹◇E!a›
4414    using "KBasic2:3"[THEN "→E", THEN "&E"(1)] by blast
4415  AOT_thus ‹∃x ◇E!x› using "∃I" by fast
4416qed
4417
4418AOT_theorem "thm-cont-e:4": ‹◇∃x ¬E!x›
4419proof (rule "CBF◇"[THEN "→E"])
4420  AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4421    using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4422  AOT_hence ‹◇◇¬E!a›
4423    using "KBasic2:3"[THEN "→E", THEN "&E"(2)] by blast
4424  AOT_hence ‹◇¬E!a›
4425    using "4◇"[THEN "→E"] by blast
4426  AOT_thus ‹∃x ◇¬E!x› using "∃I" by fast
4427qed
4428
4429AOT_theorem "thm-cont-e:5": ‹Contingent([E!])›
4430proof -
4431  AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4432    using "thm-cont-prop:2" GEN by fast
4433  AOT_hence ‹Contingent([E!]) ≡ ◇∃x E!x & ◇∃x ¬E!x›
4434    using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4435  thus ?thesis
4436    using "thm-cont-e:3" "thm-cont-e:4" "≡E"(2) "&I" by blast
4437qed
4438
4439AOT_theorem "thm-cont-e:6": ‹Contingent([E!]-)›
4440proof -
4441  AOT_have ‹∀F (Contingent([«F::<κ>»]) ≡ Contingent([F]-))›
4442    using "thm-cont-prop:3" GEN by fast
4443  AOT_hence ‹Contingent([E!]) ≡ Contingent([E!]-)›
4444    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4445  thus ?thesis using "thm-cont-e:5" "≡E" by blast
4446qed
4447
4448AOT_theorem "thm-cont-e:7": ‹∃F∃G (Contingent([«F::<κ>»]) & Contingent([G]) & F ≠ G)›
4449proof (rule "∃I")+
4450  AOT_have ‹∀F [«F::<κ>»] ≠ [F]-› using "thm-relation-negation:5" GEN by fast
4451  AOT_hence ‹[E!] ≠ [E!]-›
4452    using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4453  AOT_thus ‹Contingent([E!]) & Contingent([E!]-) & [E!] ≠ [E!]-›
4454    using "thm-cont-e:5" "thm-cont-e:6" "&I" by metis
4455next
4456  AOT_show ‹E!-↓›
4457    by (fact AOT)
4458next
4459  AOT_show ‹E!↓› by (fact "cqt:2[concrete]"[axiom_inst])
4460qed
4461
4462AOT_theorem "property-facts:1": ‹NonContingent([F]) → ¬∃G (Contingent([G]) & G = F)›
4463proof (rule "→I"; rule "raa-cor:2")
4464  AOT_assume ‹NonContingent([F])›
4465  AOT_hence 1: ‹Necessary([F]) ∨ Impossible([F])›
4466    using "contingent-properties:3"[THEN "≡dfE"] by blast
4467  AOT_assume ‹∃G (Contingent([G]) & G = F)›
4468  then AOT_obtain G where ‹Contingent([G]) & G = F› using "∃E"[rotated] by blast
4469  AOT_hence ‹Contingent([F])› using "rule=E" "&E" by blast
4470  AOT_hence ‹¬(Necessary([F]) ∨ Impossible([F]))›
4471    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4472  AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4473    using 1 "&I" by blast
4474qed
4475
4476AOT_theorem "property-facts:2": ‹Contingent([F]) → ¬∃G (NonContingent([G]) & G = F)›
4477proof (rule "→I"; rule "raa-cor:2")
4478  AOT_assume ‹Contingent([F])›
4479  AOT_hence 1: ‹¬(Necessary([F]) ∨ Impossible([F]))›
4480    using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4481  AOT_assume ‹∃G (NonContingent([G]) & G = F)›
4482  then AOT_obtain G where ‹NonContingent([G]) & G = F› using "∃E"[rotated] by blast
4483  AOT_hence ‹NonContingent([F])› using "rule=E" "&E" by blast
4484  AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4485    using "contingent-properties:3"[THEN "≡dfE"] by blast
4486  AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4487    using 1 "&I" by blast
4488qed
4489
4490AOT_theorem "property-facts:3": ‹L ≠ [L]- & L ≠ E! & L ≠ E!- & [L]- ≠ [E!]- & E! ≠ [E!]-›
4491proof -
4492  AOT_have noneqI: ‹Π ≠ Π'› if ‹φ{Π}› and ‹¬φ{Π'}› for φ and Π Π' :: ‹<κ>›
4493    apply (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
4494    using "rule=E"[where φ=φ and τ=Π and σ = Π'] that "&I" by blast
4495  AOT_have contingent_denotes: ‹Π↓› if ‹Contingent([Π])› for Π :: ‹<κ>›
4496    using that "contingent-properties:4"[THEN "≡dfE", THEN "&E"(1)] by blast
4497  AOT_have not_noncontingent_if_contingent: ‹¬NonContingent([Π])› if ‹Contingent([Π])› for Π :: ‹<κ>›
4498  proof(rule RAA(2))
4499    AOT_show ‹¬(Necessary([Π]) ∨ Impossible([Π]))›
4500      using that "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF contingent_denotes[OF that], THEN "≡E"(1)] by blast
4501  next
4502    AOT_assume ‹NonContingent([Π])›
4503    AOT_thus ‹Necessary([Π]) ∨ Impossible([Π])›
4504      using "contingent-properties:3"[THEN "≡dfE"] by blast
4505  qed
4506
4507  show ?thesis
4508  proof (safe intro!: "&I")
4509    AOT_show ‹L ≠ [L]-›
4510      apply (rule "=dfI"(2)[OF L_def])
4511       apply "cqt:2[lambda]"
4512      apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]-»"])
4513       apply (rule GEN) apply (fact AOT)
4514      by "cqt:2[lambda]"
4515  next
4516    AOT_show ‹L ≠ E!›
4517      apply (rule noneqI)
4518      using "thm-noncont-e-e:3" not_noncontingent_if_contingent[OF "thm-cont-e:5"]
4519      by auto
4520  next
4521    AOT_show ‹L ≠ E!-›
4522      apply (rule noneqI)
4523      using "thm-noncont-e-e:3" apply fast
4524      apply (rule not_noncontingent_if_contingent)
4525      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4526      using "thm-cont-prop:3" GEN apply fast
4527      using "thm-cont-e:5" by fast+
4528  next
4529    AOT_show ‹[L]- ≠ E!-›
4530      apply (rule noneqI)
4531      using "thm-noncont-e-e:4" apply fast
4532      apply (rule not_noncontingent_if_contingent)
4533      apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4534      using "thm-cont-prop:3" GEN apply fast
4535      using "thm-cont-e:5" by fast+
4536  next
4537    AOT_show ‹E! ≠ E!-›
4538      apply (rule "=dfI"(2)[OF L_def])
4539       apply "cqt:2[lambda]"
4540      apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]-»"])
4541       apply (rule GEN) apply (fact AOT)
4542      by (fact "cqt:2[concrete]"[axiom_inst])
4543  qed
4544qed
4545
4546AOT_theorem "thm-cont-propos:1": ‹NonContingent0(p) ≡ NonContingent0(((p)-))›
4547proof(rule "≡I"; rule "→I")
4548  AOT_assume ‹NonContingent0(p)›
4549  AOT_hence ‹Necessary0(p) ∨ Impossible0(p)›
4550    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4551  moreover {
4552    AOT_assume ‹Necessary0(p)›
4553    AOT_hence 1: ‹□p› using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4554    AOT_have ‹□¬((p)-)›
4555      by (AOT_subst ‹¬((p)-)› ‹p›)
4556         (auto simp add: 1 "thm-relation-negation:4")
4557    AOT_hence ‹Impossible0(((p)-))›
4558      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4559  }
4560  moreover {
4561    AOT_assume ‹Impossible0(p)›
4562    AOT_hence 1: ‹□¬p›
4563      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4564    AOT_have ‹□((p)-)›
4565      by (AOT_subst ‹((p)-)› ‹¬p›) 
4566         (auto simp: 1 "thm-relation-negation:3")
4567    AOT_hence ‹Necessary0(((p)-))›
4568      by (rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4569  }
4570  ultimately AOT_have ‹Necessary0(((p)-)) ∨ Impossible0(((p)-))›
4571    using "∨E"(1) "∨I" "→I" by metis
4572  AOT_thus ‹NonContingent0(((p)-))›
4573    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4574next
4575  AOT_assume ‹NonContingent0(((p)-))›
4576  AOT_hence ‹Necessary0(((p)-)) ∨ Impossible0(((p)-))›
4577    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4578  moreover {
4579    AOT_assume ‹Impossible0(((p)-))›
4580    AOT_hence 1: ‹□¬((p)-)›
4581      by (rule "contingent-properties:2[zero]"[THEN "≡dfE"])
4582    AOT_have ‹□p›
4583      by (AOT_subst (reverse) ‹p› ‹¬((p)-)›)
4584         (auto simp: 1 "thm-relation-negation:4")
4585    AOT_hence ‹Necessary0(p)›
4586      using "contingent-properties:1[zero]"[THEN "≡dfI"] by blast
4587  }
4588  moreover {
4589    AOT_assume ‹Necessary0(((p)-))›
4590    AOT_hence 1: ‹□((p)-)›
4591      by (rule "contingent-properties:1[zero]"[THEN "≡dfE"])
4592    AOT_have ‹□¬p›
4593      by (AOT_subst (reverse) ‹¬p› ‹((p)-)›)
4594         (auto simp: 1 "thm-relation-negation:3")
4595    AOT_hence ‹Impossible0(p)›
4596      by (rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4597  }
4598  ultimately AOT_have ‹Necessary0(p) ∨ Impossible0(p)›
4599    using "∨E"(1) "∨I" "→I" by metis
4600  AOT_thus ‹NonContingent0(p)›
4601    using "contingent-properties:3[zero]"[THEN "≡dfI"] by blast
4602qed
4603
4604AOT_theorem "thm-cont-propos:2": ‹Contingent0(φ) ≡ ◇φ & ◇¬φ›
4605proof -
4606  AOT_have ‹Contingent0(φ) ≡ ¬(Necessary0(φ) ∨ Impossible0(φ))›
4607    using "contingent-properties:4[zero]"[THEN "≡Df"] by simp
4608  also AOT_have ‹… ≡ ¬Necessary0(φ) & ¬Impossible0(φ)›
4609    by (fact AOT)
4610  also AOT_have ‹… ≡ ¬Impossible0(φ) & ¬Necessary0(φ)›
4611    by (fact AOT)
4612  also AOT_have ‹… ≡ ◇φ & ◇¬φ›
4613    apply (AOT_subst ‹◇φ› ‹¬□¬φ›)
4614     apply (simp add: "conventions:5" "≡Df")
4615    apply (AOT_subst ‹Impossible0(φ)› ‹□¬φ›)
4616     apply (simp add: "contingent-properties:2[zero]" "≡Df")
4617    apply (AOT_subst (reverse) ‹◇¬φ› ‹¬□φ›)
4618     apply (simp add: "KBasic:11")
4619    apply (AOT_subst ‹Necessary0(φ)› ‹□φ›)
4620     apply (simp add: "contingent-properties:1[zero]" "≡Df")
4621    by (simp add: "oth-class-taut:3:a")
4622  finally show ?thesis.
4623qed
4624
4625AOT_theorem "thm-cont-propos:3": ‹Contingent0(p) ≡ Contingent0(((p)-))›
4626proof -
4627  AOT_have ‹Contingent0(p) ≡ ◇p & ◇¬p› using "thm-cont-propos:2".
4628  also AOT_have ‹… ≡ ◇¬p & ◇p› by (fact AOT)
4629  also AOT_have ‹… ≡ ◇((p)-) & ◇p›
4630    by (AOT_subst ‹((p)-)› ‹¬p›)
4631       (auto simp: "thm-relation-negation:3" "oth-class-taut:3:a")
4632  also AOT_have ‹… ≡ ◇((p)-) & ◇¬((p)-)›
4633    by (AOT_subst ‹¬((p)-)› ‹p›)
4634       (auto simp: "thm-relation-negation:4" "oth-class-taut:3:a")
4635  also AOT_have ‹… ≡ Contingent0(((p)-))›
4636    using "thm-cont-propos:2"[symmetric] by blast
4637  finally show ?thesis.
4638qed
4639
4640AOT_define noncontingent_prop :: ‹φ› ("p0")
4641  p0_def: "(p0) =df (∀x (E!x → E!x))"
4642
4643AOT_theorem "thm-noncont-propos:1":  ‹Necessary0((p0))›
4644proof(rule "contingent-properties:1[zero]"[THEN "≡dfI"])
4645  AOT_show ‹□(p0)›
4646    apply (rule "=dfI"(2)[OF p0_def])
4647    using "log-prop-prop:2" apply simp
4648    using "if-p-then-p" RN GEN by fast
4649qed
4650
4651AOT_theorem "thm-noncont-propos:2": ‹Impossible0(((p0)-))›
4652proof(rule "contingent-properties:2[zero]"[THEN "≡dfI"])
4653  AOT_show ‹□¬((p0)-)›
4654    apply (AOT_subst ‹((p0)-)› ‹¬p0›)
4655    using "thm-relation-negation:3" GEN "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4656    apply (AOT_subst (reverse) ‹¬¬p0› ‹p0›)
4657     apply (simp add: "oth-class-taut:3:b")
4658    apply (rule "=dfI"(2)[OF p0_def])
4659    using "log-prop-prop:2" apply simp
4660    using "if-p-then-p" RN GEN by fast
4661qed
4662
4663AOT_theorem "thm-noncont-propos:3": ‹NonContingent0((p0))›
4664  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4665  using "thm-noncont-propos:1" "∨I" by blast
4666
4667AOT_theorem "thm-noncont-propos:4": ‹NonContingent0(((p0)-))›
4668  apply(rule "contingent-properties:3[zero]"[THEN "≡dfI"])
4669  using "thm-noncont-propos:2" "∨I" by blast
4670
4671AOT_theorem "thm-noncont-propos:5": ‹∃p∃q (NonContingent0((p)) & NonContingent0((q)) & p ≠ q)›
4672proof(rule "∃I")+
4673  AOT_have 0: ‹φ ≠ (φ)-› for φ
4674    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4675  AOT_thus ‹NonContingent0((p0)) & NonContingent0(((p0)-)) & (p0) ≠ (p0)-›
4676    using "thm-noncont-propos:3" "thm-noncont-propos:4" "&I" by auto
4677qed(auto simp: "log-prop-prop:2")
4678
4679AOT_act_theorem "no-cnac": ‹¬∃x(E!x & ¬𝒜E!x)›
4680proof(rule "raa-cor:2")
4681  AOT_assume ‹∃x(E!x & ¬𝒜E!x)›
4682  then AOT_obtain a where a: ‹E!a & ¬𝒜E!a›
4683    using "∃E"[rotated] by blast
4684  AOT_hence ‹𝒜¬E!a› using "&E" "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
4685  AOT_hence ‹¬E!a› using "logic-actual"[act_axiom_inst, THEN "→E"] by blast
4686  AOT_hence ‹E!a & ¬E!a› using a "&E" "&I" by blast
4687  AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
4688qed
4689
4690AOT_theorem "pos-not-pna:1": ‹¬𝒜∃x (E!x & ¬𝒜E!x)›
4691proof(rule "raa-cor:2")
4692  AOT_assume ‹𝒜∃x (E!x & ¬𝒜E!x)›
4693  AOT_hence ‹∃x 𝒜(E!x & ¬𝒜E!x)›
4694    using "Act-Basic:10"[THEN "≡E"(1)] by blast
4695  then AOT_obtain a where ‹𝒜(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
4696  AOT_hence 1: ‹𝒜E!a & 𝒜¬𝒜E!a› using "Act-Basic:2"[THEN "≡E"(1)] by blast
4697  AOT_hence ‹¬𝒜𝒜E!a› using "&E"(2) "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by blast
4698  AOT_hence ‹¬𝒜E!a› using "logic-actual-nec:4"[axiom_inst, THEN "≡E"(1)] RAA by blast
4699  AOT_thus ‹p & ¬p› for p using 1[THEN "&E"(1)] "&I" "raa-cor:1" by blast
4700qed
4701
4702AOT_theorem "pos-not-pna:2": ‹◇¬∃x(E!x & ¬𝒜E!x)›
4703proof (rule RAA(1))
4704  AOT_show ‹¬𝒜∃x (E!x & ¬𝒜E!x)› using "pos-not-pna:1" by blast
4705next
4706  AOT_assume ‹¬◇¬∃x (E!x & ¬𝒜E!x)›
4707  AOT_hence ‹□∃x (E!x & ¬𝒜E!x)›
4708    using "KBasic:12"[THEN "≡E"(2)] by blast
4709  AOT_thus ‹𝒜∃x (E!x & ¬𝒜E!x)›
4710    using "nec-imp-act"[THEN "→E"] by blast
4711qed
4712
4713AOT_theorem "pos-not-pna:3": ‹∃x (◇E!x & ¬𝒜E!x)›
4714proof -
4715  AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)›
4716    using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] "∃E"[rotated] by blast
4717  AOT_hence θ: ‹◇E!a› and ξ: ‹◇¬𝒜E!a› using "KBasic2:3"[THEN "→E"] "&E" by blast+
4718  AOT_have ‹¬□𝒜E!a› using ξ "KBasic:11"[THEN "≡E"(2)] by blast
4719  AOT_hence ‹¬𝒜E!a› using "Act-Basic:6"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)] by blast
4720  AOT_hence ‹◇E!a & ¬𝒜E!a› using θ "&I" by blast
4721  thus ?thesis using "∃I" by fast
4722qed
4723
4724AOT_define contingent_prop :: φ ("q0")
4725  q0_def: ‹(q0) =df (∃x (E!x & ¬𝒜E!x))›
4726
4727AOT_theorem q0_prop: ‹◇q0 & ◇¬q0›
4728  apply (rule "=dfI"(2)[OF q0_def])
4729  apply (fact "log-prop-prop:2")
4730  apply (rule "&I")
4731   apply (fact "qml:4"[axiom_inst])
4732  by (fact "pos-not-pna:2")
4733
4734AOT_theorem "basic-prop:1": ‹Contingent0((q0))›
4735proof(rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4736  AOT_have ‹¬Necessary0((q0)) & ¬Impossible0((q0))›
4737  proof (rule "&I"; rule "=dfI"(2)[OF q0_def]; (rule "log-prop-prop:2" | rule "raa-cor:2"))
4738    AOT_assume ‹Necessary0(∃x (E!x & ¬𝒜E!x))›
4739    AOT_hence ‹□∃x (E!x & ¬𝒜E!x)›
4740      using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4741    AOT_hence ‹𝒜∃x (E!x & ¬𝒜E!x)›
4742      using "Act-Basic:8"[THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
4743    AOT_thus ‹𝒜∃x (E!x & ¬𝒜E!x) & ¬𝒜∃x (E!x & ¬𝒜E!x)›
4744      using "pos-not-pna:1" "&I" by blast
4745  next
4746    AOT_assume ‹Impossible0(∃x (E!x & ¬𝒜E!x))›
4747    AOT_hence ‹□¬(∃x (E!x & ¬𝒜E!x))›
4748      using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4749    AOT_hence ‹¬◇(∃x (E!x & ¬𝒜E!x))› using "KBasic2:1"[THEN "≡E"(1)] by blast
4750    AOT_thus ‹◇(∃x (E!x & ¬𝒜E!x)) & ¬◇(∃x (E!x & ¬𝒜E!x))›
4751      using "qml:4"[axiom_inst] "&I" by blast
4752  qed
4753  AOT_thus ‹¬(Necessary0((q0)) ∨ Impossible0((q0)))›
4754    using "oth-class-taut:5:d" "≡E"(2) by blast
4755qed
4756
4757AOT_theorem "basic-prop:2": ‹∃p Contingent0((p))›
4758  using "∃I"(1)[rotated, OF "log-prop-prop:2"] "basic-prop:1" by blast
4759
4760AOT_theorem "basic-prop:3": ‹Contingent0(((q0)-))›
4761  apply (AOT_subst ‹((q0)-)› ‹¬q0›)
4762   apply (insert "thm-relation-negation:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"]; fast)
4763  apply (rule "contingent-properties:4[zero]"[THEN "≡dfI"])
4764  apply (rule "oth-class-taut:5:d"[THEN "≡E"(2)])
4765  apply (rule "&I")
4766   apply (rule "contingent-properties:1[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4767   apply (rule "conventions:5"[THEN "≡dfE"])
4768   apply (rule "=dfE"(2)[OF q0_def])
4769    apply (rule "log-prop-prop:2")
4770   apply (rule q0_prop[THEN "&E"(1)])
4771  apply (rule "contingent-properties:2[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4772  apply (rule "conventions:5"[THEN "≡dfE"])
4773  by (rule q0_prop[THEN "&E"(2)])
4774
4775AOT_theorem "basic-prop:4": ‹∃p∃q (p ≠ q & Contingent0(p) & Contingent0(q))›
4776proof(rule "∃I")+
4777  AOT_have 0: ‹φ ≠ (φ)-› for φ
4778    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4779  AOT_show ‹(q0) ≠ (q0)- & Contingent0(q0) & Contingent0(((q0)-))›
4780    using "basic-prop:1" "basic-prop:3" "&I" 0 by presburger
4781qed(auto simp: "log-prop-prop:2")
4782
4783AOT_theorem "proposition-facts:1": ‹NonContingent0(p) → ¬∃q (Contingent0(q) & q = p)›
4784proof(rule "→I"; rule "raa-cor:2")
4785  AOT_assume ‹NonContingent0(p)›
4786  AOT_hence 1: ‹Necessary0(p) ∨ Impossible0(p)›
4787    using "contingent-properties:3[zero]"[THEN "≡dfE"] by blast
4788  AOT_assume ‹∃q (Contingent0(q) & q = p)›
4789  then AOT_obtain q where ‹Contingent0(q) & q = p› using "∃E"[rotated] by blast
4790  AOT_hence ‹Contingent0(p)› using "rule=E" "&E" by fast
4791  AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4792    using "contingent-properties:4[zero]"[THEN "≡dfE"] 1 "&I" by blast
4793qed
4794
4795AOT_theorem "proposition-facts:2": ‹Contingent0(p) → ¬∃q (NonContingent0(q) & q = p)›
4796proof(rule "→I"; rule "raa-cor:2")
4797  AOT_assume ‹Contingent0(p)›
4798  AOT_hence 1: ‹¬(Necessary0(p) ∨ Impossible0(p))›
4799    using "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4800  AOT_assume ‹∃q (NonContingent0(q) & q = p)›
4801  then AOT_obtain q where ‹NonContingent0(q) & q = p› using "∃E"[rotated] by blast
4802  AOT_hence ‹NonContingent0(p)› using "rule=E" "&E" by fast
4803  AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4804    using "contingent-properties:3[zero]"[THEN "≡dfE"] 1 "&I" by blast
4805qed
4806
4807AOT_theorem "proposition-facts:3": ‹(p0) ≠ (p0)- & (p0) ≠ (q0) & (p0) ≠ (q0)- & (p0)- ≠ (q0)- & (q0) ≠ (q0)-›
4808proof -
4809  {
4810    fix χ φ ψ
4811    AOT_assume ‹χ{φ}›
4812    moreover AOT_assume ‹¬χ{ψ}›
4813    ultimately AOT_have ‹¬(χ{φ} ≡ χ{ψ})›
4814      using RAA "≡E" by metis
4815    moreover {
4816      AOT_have ‹∀p∀q ((¬(χ{p} ≡ χ{q})) → p ≠ q)›
4817        by (rule "∀I"; rule "∀I"; rule "pos-not-equiv-ne:4[zero]")
4818      AOT_hence ‹((¬(χ{φ} ≡ χ{ψ})) → φ ≠ ψ)›
4819        using "∀E" "log-prop-prop:2" by blast
4820    }
4821    ultimately AOT_have ‹φ ≠ ψ›
4822      using "→E" by blast
4823  } note 0 = this
4824  AOT_have contingent_neg: ‹Contingent0(φ) ≡ Contingent0(((φ)-))› for φ
4825    using "thm-cont-propos:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4826  AOT_have not_noncontingent_if_contingent: ‹¬NonContingent0(φ)› if ‹Contingent0(φ)› for φ
4827    apply (rule "contingent-properties:3[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4828    using that "contingent-properties:4[zero]"[THEN "≡dfE"] by blast
4829  show ?thesis
4830    apply (rule "&I")+
4831    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4832       apply (rule 0)
4833    using "thm-noncont-propos:3" apply fast
4834       apply (rule not_noncontingent_if_contingent)
4835       apply (fact AOT)
4836      apply (rule 0)
4837    apply (rule "thm-noncont-propos:3")
4838      apply (rule not_noncontingent_if_contingent)
4839      apply (rule contingent_neg[THEN "≡E"(1)])
4840      apply (fact AOT)
4841     apply (rule 0)
4842    apply (rule "thm-noncont-propos:4")
4843      apply (rule not_noncontingent_if_contingent)
4844      apply (rule contingent_neg[THEN "≡E"(1)])
4845     apply (fact AOT)
4846    using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4847qed
4848
4849AOT_define ContingentlyTrue :: ‹φ ⇒ φ› ("ContingentlyTrue'(_')")
4850  "cont-tf:1": ‹ContingentlyTrue(p) ≡df p & ◇¬p›
4851
4852AOT_define ContingentlyFalse :: ‹φ ⇒ φ› ("ContingentlyFalse'(_')")
4853  "cont-tf:2": ‹ContingentlyFalse(p) ≡df ¬p & ◇p›
4854
4855AOT_theorem "cont-true-cont:1": ‹ContingentlyTrue((p)) → Contingent0((p))›
4856proof(rule "→I")
4857  AOT_assume ‹ContingentlyTrue((p))›
4858  AOT_hence 1: ‹p› and 2: ‹◇¬p› using "cont-tf:1"[THEN "≡dfE"] "&E" by blast+
4859  AOT_have ‹¬Necessary0((p))›
4860    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4861    using 2 "KBasic:11"[THEN "≡E"(2)] by blast
4862  moreover AOT_have ‹¬Impossible0((p))›
4863    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4864    apply (rule "conventions:5"[THEN "≡dfE"])
4865    using "T◇"[THEN "→E", OF 1].
4866  ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4867    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4868  AOT_thus ‹Contingent0((p))›
4869    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4870qed
4871
4872AOT_theorem "cont-true-cont:2": ‹ContingentlyFalse((p)) → Contingent0((p))›
4873proof(rule "→I")
4874  AOT_assume ‹ContingentlyFalse((p))›
4875  AOT_hence 1: ‹¬p› and 2: ‹◇p› using "cont-tf:2"[THEN "≡dfE"] "&E" by blast+
4876  AOT_have ‹¬Necessary0((p))›
4877    apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4878    using "KBasic:11"[THEN "≡E"(2)] "T◇"[THEN "→E", OF 1] by blast
4879  moreover AOT_have ‹¬Impossible0((p))›
4880    apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4881    apply (rule "conventions:5"[THEN "≡dfE"])
4882    using 2.
4883  ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4884    using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4885  AOT_thus ‹Contingent0((p))›
4886    using "contingent-properties:4[zero]"[THEN "≡dfI"] by blast
4887qed
4888
4889AOT_theorem "cont-true-cont:3": ‹ContingentlyTrue((p)) ≡ ContingentlyFalse(((p)-))›
4890proof(rule "≡I"; rule "→I")
4891  AOT_assume ‹ContingentlyTrue((p))›
4892  AOT_hence 0: ‹p & ◇¬p› using "cont-tf:1"[THEN "≡dfE"] by blast
4893  AOT_have 1: ‹ContingentlyFalse(¬p)›
4894    apply (rule "cont-tf:2"[THEN "≡dfI"])
4895    apply (AOT_subst (reverse) ‹¬¬p› p)
4896    by (auto simp: "oth-class-taut:3:b" 0)
4897  AOT_show ‹ContingentlyFalse(((p)-))›
4898    apply (AOT_subst ‹((p)-)› ‹¬p›)
4899    by (auto simp: "thm-relation-negation:3" 1)
4900next
4901  AOT_assume 1: ‹ContingentlyFalse(((p)-))›
4902  AOT_have ‹ContingentlyFalse(¬p)›
4903    by (AOT_subst (reverse) ‹¬p› ‹((p)-)›)
4904       (auto simp: "thm-relation-negation:3" 1)
4905  AOT_hence ‹¬¬p & ◇¬p› using "cont-tf:2"[THEN "≡dfE"] by blast
4906  AOT_hence ‹p & ◇¬p›
4907    using "&I" "&E" "useful-tautologies:1"[THEN "→E"] by metis
4908  AOT_thus ‹ContingentlyTrue((p))›
4909    using "cont-tf:1"[THEN "≡dfI"] by blast
4910qed
4911
4912AOT_theorem "cont-true-cont:4": ‹ContingentlyFalse((p)) ≡ ContingentlyTrue(((p)-))›
4913proof(rule "≡I"; rule "→I")
4914  AOT_assume ‹ContingentlyFalse(p)›
4915  AOT_hence 0: ‹¬p & ◇p›
4916    using "cont-tf:2"[THEN "≡dfE"] by blast
4917  AOT_have ‹¬p & ◇¬¬p›
4918    by (AOT_subst (reverse) ‹¬¬p› p)
4919       (auto simp: "oth-class-taut:3:b" 0)
4920  AOT_hence 1: ‹ContingentlyTrue(¬p)›
4921    by (rule "cont-tf:1"[THEN "≡dfI"])
4922  AOT_show ‹ContingentlyTrue(((p)-))›
4923    by (AOT_subst ‹((p)-)› ‹¬p›)
4924       (auto simp: "thm-relation-negation:3" 1)
4925next
4926  AOT_assume 1: ‹ContingentlyTrue(((p)-))›
4927  AOT_have ‹ContingentlyTrue(¬p)›
4928    by (AOT_subst (reverse) ‹¬p› ‹((p)-)›)
4929       (auto simp add: "thm-relation-negation:3" 1)
4930  AOT_hence 2: ‹¬p & ◇¬¬p› using "cont-tf:1"[THEN "≡dfE"] by blast
4931  AOT_have ‹◇p›
4932    by (AOT_subst p ‹¬¬p›)
4933       (auto simp add: "oth-class-taut:3:b" 2[THEN "&E"(2)])
4934  AOT_hence ‹¬p & ◇p› using 2[THEN "&E"(1)] "&I" by blast
4935  AOT_thus ‹ContingentlyFalse(p)›
4936    by (rule "cont-tf:2"[THEN "≡dfI"])
4937qed
4938
4939AOT_theorem "cont-true-cont:5": ‹(ContingentlyTrue((p)) & Necessary0((q))) → p ≠ q›
4940proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4941  AOT_assume ‹ContingentlyTrue((p))›
4942  AOT_hence ‹◇¬p›
4943    using "cont-tf:1"[THEN "≡dfE"] "&E" by blast
4944  AOT_hence 0: ‹¬□p› using "KBasic:11"[THEN "≡E"(2)] by blast
4945  AOT_assume ‹Necessary0((q))›
4946  moreover AOT_assume ‹¬(p ≠ q)›
4947  AOT_hence ‹p = q›
4948    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4949          "useful-tautologies:1"[THEN "→E"] by blast
4950  ultimately AOT_have ‹Necessary0((p))› using "rule=E" id_sym by blast
4951  AOT_hence ‹□p›
4952    using "contingent-properties:1[zero]"[THEN "≡dfE"] by blast
4953  AOT_thus ‹□p & ¬□p› using 0 "&I" by blast
4954qed
4955
4956AOT_theorem "cont-true-cont:6": ‹(ContingentlyFalse((p)) & Impossible0((q))) → p ≠ q›
4957proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4958  AOT_assume ‹ContingentlyFalse((p))›
4959  AOT_hence ‹◇p›
4960    using "cont-tf:2"[THEN "≡dfE"] "&E" by blast
4961  AOT_hence 1: ‹¬□¬p›
4962    using "conventions:5"[THEN "≡dfE"] by blast
4963  AOT_assume ‹Impossible0((q))›
4964  moreover AOT_assume ‹¬(p ≠ q)›
4965  AOT_hence ‹p = q›
4966    using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4967          "useful-tautologies:1"[THEN "→E"] by blast
4968  ultimately AOT_have ‹Impossible0((p))› using "rule=E" id_sym by blast
4969  AOT_hence ‹□¬p›
4970    using "contingent-properties:2[zero]"[THEN "≡dfE"] by blast
4971  AOT_thus ‹□¬p & ¬□¬p› using 1 "&I" by blast
4972qed
4973
4974AOT_act_theorem "q0cf:1": ‹ContingentlyFalse(q0)›
4975  apply (rule "cont-tf:2"[THEN "≡dfI"])
4976  apply (rule "=dfI"(2)[OF q0_def])
4977   apply (fact "log-prop-prop:2")
4978  apply (rule "&I")
4979   apply (fact "no-cnac")
4980  by (fact "qml:4"[axiom_inst])
4981
4982AOT_act_theorem "q0cf:2": ‹ContingentlyTrue(((q0)-))›
4983  apply (rule "cont-tf:1"[THEN "≡dfI"])
4984  apply (rule "=dfI"(2)[OF q0_def])
4985   apply (fact "log-prop-prop:2")
4986  apply (rule "&I")
4987     apply (rule "thm-relation-negation:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(2)])
4988     apply (fact "no-cnac")
4989    apply (rule "rule=E"[rotated, OF "thm-relation-negation:7"[unvarify p, OF "log-prop-prop:2", THEN id_sym]])
4990  apply (AOT_subst (reverse) ‹¬¬(∃x  (E!x & ¬𝒜E!x))› ‹∃x (E!x & ¬𝒜E!x)›)
4991  by (auto simp: "oth-class-taut:3:b" "qml:4"[axiom_inst])
4992
4993(* TODO: q0cf-rem skipped for now *)
4994
4995AOT_theorem "cont-tf-thm:1": ‹∃p ContingentlyTrue((p))›
4996proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
4997  AOT_assume ‹q0›
4998  AOT_hence ‹q0 & ◇¬q0› using q0_prop[THEN "&E"(2)] "&I" by blast
4999  AOT_thus ‹ContingentlyTrue(q0)›
5000    by (rule "cont-tf:1"[THEN "≡dfI"])
5001next
5002  AOT_assume ‹¬q0›
5003  AOT_hence ‹¬q0 & ◇q0› using q0_prop[THEN "&E"(1)] "&I" by blast
5004  AOT_hence ‹ContingentlyFalse(q0)›
5005    by (rule "cont-tf:2"[THEN "≡dfI"])
5006  AOT_thus ‹ContingentlyTrue(((q0)-))›
5007    by (rule "cont-true-cont:4"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
5008qed(auto simp: "log-prop-prop:2")
5009
5010
5011AOT_theorem "cont-tf-thm:2": ‹∃p ContingentlyFalse((p))›
5012proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
5013  AOT_assume ‹q0›
5014  AOT_hence ‹q0 & ◇¬q0› using q0_prop[THEN "&E"(2)] "&I" by blast
5015  AOT_hence ‹ContingentlyTrue(q0)›
5016    by (rule "cont-tf:1"[THEN "≡dfI"])
5017  AOT_thus ‹ContingentlyFalse(((q0)-))›
5018    by (rule "cont-true-cont:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
5019next
5020  AOT_assume ‹¬q0›
5021  AOT_hence ‹¬q0 & ◇q0› using q0_prop[THEN "&E"(1)] "&I" by blast
5022  AOT_thus ‹ContingentlyFalse(q0)›
5023    by (rule "cont-tf:2"[THEN "≡dfI"])
5024qed(auto simp: "log-prop-prop:2")
5025
5026(* TODO: inspect modally strict subproof involving obtained variable *)
5027AOT_theorem "property-facts1:1": ‹∃F∃x ([F]x & ◇¬[F]x)›
5028proof -
5029  fix x
5030  AOT_obtain p1 where ‹ContingentlyTrue((p1))›
5031    using "cont-tf-thm:1" "∃E"[rotated] by blast
5032  AOT_hence 1: ‹p1 & ◇¬p1› using "cont-tf:1"[THEN "≡dfE"] by blast
5033  AOT_modally_strict {
5034    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
5035      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5036    AOT_hence ‹for arbitrary p: ⊢□ □ ([λz p]x ≡ p)›
5037      by (rule RN)
5038    AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
5039    AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by fast
5040  } note 2 = this
5041  AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by blast
5042  AOT_hence ‹[λz p1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
5043  moreover AOT_have ‹◇¬[λz p1]x›
5044    apply (AOT_subst_using subst: 2[THEN "qml:2"[axiom_inst, THEN "→E"]])
5045    using 1[THEN "&E"(2)] by blast
5046  ultimately AOT_have ‹[λz p1]x & ◇¬[λz p1]x› using "&I" by blast
5047  AOT_hence ‹∃x ([λz p1]x & ◇¬[λz p1]x)› using "∃I"(2) by fast
5048  moreover AOT_have ‹[λz p1]↓› by "cqt:2[lambda]"
5049  ultimately AOT_show ‹∃F∃x ([F]x & ◇¬[F]x)› by (rule "∃I"(1))
5050qed
5051
5052(* TODO: inspect modally strict subproof involving obtained variable *)
5053AOT_theorem "property-facts1:2": ‹∃F∃x (¬[F]x & ◇[F]x)›
5054proof -
5055  fix x
5056  AOT_obtain p1 where ‹ContingentlyFalse((p1))›
5057    using "cont-tf-thm:2" "∃E"[rotated] by blast
5058  AOT_hence 1: ‹¬p1 & ◇p1› using "cont-tf:2"[THEN "≡dfE"] by blast
5059  AOT_modally_strict {
5060    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
5061      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5062    AOT_hence ‹for arbitrary p: ⊢□ (¬[λz p]x ≡ ¬p)›
5063      using "oth-class-taut:4:b" "≡E" by blast
5064    AOT_hence ‹for arbitrary p: ⊢□ □(¬[λz p]x ≡ ¬p)›
5065      by (rule RN)
5066    AOT_hence ‹∀p □(¬[λz p]x ≡ ¬p)› using GEN by fast
5067    AOT_hence ‹□(¬[λz p1]x ≡ ¬p1)› using "∀E" by fast
5068  } note 2 = this
5069  AOT_hence ‹□(¬[λz p1]x ≡ ¬p1)› using "∀E" by blast
5070  AOT_hence 3: ‹¬[λz p1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
5071  AOT_modally_strict {
5072    AOT_have ‹for arbitrary p: ⊢□ ([λz p]x ≡ p)›
5073      by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5074    AOT_hence ‹for arbitrary p: ⊢□ □([λz p]x ≡ p)›
5075      by (rule RN)
5076    AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
5077    AOT_hence ‹□([λz p1]x ≡ p1)› using "∀E" by fast
5078  } note 4 = this
5079  AOT_have ‹◇[λz p1]x›
5080    apply (AOT_subst_using subst: 4[THEN "qml:2"[axiom_inst, THEN "→E"]])
5081    using 1[THEN "&E"(2)] by blast
5082  AOT_hence ‹¬[λz p1]x & ◇[λz p1]x› using 3 "&I" by blast
5083  AOT_hence ‹∃x (¬[λz p1]x & ◇[λz p1]x)› using "∃I"(2) by fast
5084  moreover AOT_have ‹[λz p1]↓› by "cqt:2[lambda]"
5085  ultimately AOT_show ‹∃F∃x (¬[F]x & ◇[F]x)› by (rule "∃I"(1))
5086qed
5087
5088context
5089begin
5090
5091private AOT_lemma eqnotnec_123_Aux_ζ: ‹[L]x ≡ (E!x → E!x)›
5092    apply (rule "=dfI"(2)[OF L_def])
5093     apply "cqt:2[lambda]"
5094    apply (rule "beta-C-meta"[THEN "→E"])
5095  by "cqt:2[lambda]"
5096
5097private AOT_lemma eqnotnec_123_Aux_ω: ‹[λz φ]x ≡ φ›
5098    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5099
5100private AOT_lemma eqnotnec_123_Aux_θ: ‹φ ≡ ∀x([L]x ≡ [λz φ]x)›
5101proof(rule "≡I"; rule "→I"; (rule "∀I")?)
5102  fix x
5103  AOT_assume 1: ‹φ›
5104  AOT_have ‹[L]x ≡ (E!x → E!x)› using eqnotnec_123_Aux_ζ.
5105  also AOT_have ‹… ≡ φ›
5106    using "if-p-then-p" 1 "≡I" "→I" by simp
5107  also AOT_have ‹… ≡ [λz φ]x›
5108    using "Commutativity of ≡"[THEN "≡E"(1)] eqnotnec_123_Aux_ω by blast
5109  finally AOT_show ‹[L]x ≡ [λz φ]x›.
5110next
5111  fix x
5112  AOT_assume ‹∀x([L]x ≡ [λz φ]x)›
5113  AOT_hence ‹[L]x ≡ [λz φ]x› using "∀E" by blast
5114  also AOT_have ‹… ≡ φ› using eqnotnec_123_Aux_ω.
5115  finally AOT_have ‹φ ≡ [L]x› using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5116  also AOT_have ‹… ≡ E!x → E!x› using eqnotnec_123_Aux_ζ.
5117  finally AOT_show ‹φ› using "≡E" "if-p-then-p" by fast
5118qed
5119private lemmas eqnotnec_123_Aux_ξ =  eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5120                      THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5121                      THEN "RM◇"]
5122private lemmas eqnotnec_123_Aux_ξ' = eqnotnec_123_Aux_θ[THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)], THEN "RM◇"]
5123
5124AOT_theorem "eqnotnec:1": ‹∃F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5125proof-
5126  AOT_obtain p1 where ‹ContingentlyTrue(p1)› using "cont-tf-thm:1" "∃E"[rotated] by blast
5127  AOT_hence ‹p1 & ◇¬p1› using "cont-tf:1"[THEN "≡dfE"] by blast
5128  AOT_hence ‹∀x ([L]x ≡ [λz p1]x) & ◇¬∀x([L]x ≡ [λz p1]x)›
5129    apply - apply (rule "&I")
5130    using "&E" eqnotnec_123_Aux_θ[THEN "≡E"(1)] eqnotnec_123_Aux_ξ "→E" by fast+
5131  AOT_hence ‹∃G (∀x([L]x ≡ [G]x) & ◇¬∀x([L]x ≡ [G]x))›
5132    by (rule "∃I") "cqt:2[lambda]"
5133  AOT_thus ‹∃F∃G (∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5134    apply (rule "∃I")
5135    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5136qed
5137
5138AOT_theorem "eqnotnec:2": ‹∃F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5139proof-
5140  AOT_obtain p1 where ‹ContingentlyFalse(p1)› using "cont-tf-thm:2" "∃E"[rotated] by blast
5141  AOT_hence ‹¬p1 & ◇p1› using "cont-tf:2"[THEN "≡dfE"] by blast
5142  AOT_hence ‹¬∀x ([L]x ≡ [λz p1]x) & ◇∀x([L]x ≡ [λz p1]x)›
5143    apply - apply (rule "&I")
5144    using "&E" eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)] eqnotnec_123_Aux_ξ' "→E" by fast+
5145  AOT_hence ‹∃G (¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5146    by (rule "∃I") "cqt:2[lambda]"
5147  AOT_thus ‹∃F∃G (¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5148    apply (rule "∃I")
5149    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5150qed
5151
5152AOT_theorem "eqnotnec:3": ‹∃F∃G(𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5153proof-
5154  AOT_have ‹¬𝒜q0›
5155    apply (rule "=dfI"(2)[OF q0_def])
5156     apply (fact "log-prop-prop:2")
5157    by (fact AOT)
5158  AOT_hence ‹𝒜¬q0›
5159    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5160  AOT_hence ‹𝒜¬∀x ([L]x ≡ [λz q0]x)›
5161    using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5162            THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5163            THEN "RA[2]", THEN "act-cond"[THEN "→E"], THEN "→E"] by blast
5164  moreover AOT_have ‹◇∀x ([L]x ≡ [λz q0]x)› using eqnotnec_123_Aux_ξ'[THEN "→E"] q0_prop[THEN "&E"(1)] by blast
5165  ultimately AOT_have ‹𝒜¬∀x ([L]x ≡ [λz q0]x) & ◇∀x ([L]x ≡ [λz q0]x)› using "&I" by blast
5166  AOT_hence ‹∃G (𝒜¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5167    by (rule "∃I") "cqt:2[lambda]"
5168  AOT_thus ‹∃F∃G (𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5169    apply (rule "∃I")
5170    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5171qed
5172
5173end
5174
5175(* TODO[IMPORTANT]: proof of 219.4 ζ: appeal to (159.2) requires a theorem, but the result has local
5176   assumptions! *)
5177AOT_theorem "eqnotnec:4": ‹∀F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5178proof(rule GEN)
5179  fix F
5180
5181  AOT_have Aux_A: ‹⊢□ ψ → ∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5182  proof(rule "→I"; rule GEN)
5183    AOT_modally_strict {
5184    fix x
5185    AOT_assume 0: ‹ψ›
5186    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5187      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5188    also AOT_have ‹... ≡ [F]x›
5189      apply (rule "≡I"; rule "→I")
5190      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5191      using 0 "&I" by blast
5192    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5193      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5194    }
5195  qed
5196
5197  AOT_have Aux_B: ‹⊢□ ψ → ∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5198  proof (rule "→I"; rule GEN)
5199    AOT_modally_strict {
5200      fix x
5201      AOT_assume 0: ‹ψ›
5202      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5203        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5204      also AOT_have ‹... ≡ [F]x›
5205        apply (rule "≡I"; rule "→I")
5206        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5207        apply (rule "∨I"(1)) using 0 "&I" by blast
5208      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5209        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5210    }
5211  qed
5212
5213  AOT_have Aux_C: ‹⊢□ ◇¬ψ → ◇¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5214  proof(rule "RM◇"; rule "→I"; rule "raa-cor:2")
5215  AOT_modally_strict {
5216      AOT_assume 0: ‹¬ψ›
5217      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5218      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5219      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5220          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5221      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5222        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5223      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5224        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5225      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5226      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5227      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5228    }
5229  qed
5230
5231  AOT_have Aux_D: ‹□∀z ([F]z ≡ [λz [F]z & ψ]z) → (◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5232  proof (rule "→I")
5233    AOT_assume A: ‹□∀z([F]z ≡ [λz [F]z & ψ]z)›
5234    AOT_show ‹◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5235    proof(rule "≡I"; rule "KBasic:13"[THEN "→E"];
5236          rule "RN[prem]"[where Γ="{«∀z([F]z ≡ [λz [F]z & ψ]z)»}", simplified];
5237          (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5238      AOT_modally_strict {
5239        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5240        AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5241        AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5242        AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5243        AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5244        AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5245      }
5246    next
5247      AOT_modally_strict {
5248        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5249        AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5250        AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5251        AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5252        AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5253        AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5254      }
5255    qed(auto simp: A)
5256  qed
5257
5258  AOT_obtain p1 where p1_prop: ‹p1 & ◇¬p1› using "cont-tf-thm:1" "∃E"[rotated] "cont-tf:1"[THEN "≡dfE"] by blast
5259  {
5260    AOT_assume 1: ‹□∀x([F]x ≡ [λz [F]z & p1]x)›
5261    AOT_have 2: ‹∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5262      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5263    AOT_have ‹◇¬∀x([λz [F]z & p1]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5264      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5265    AOT_hence 3: ‹◇¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5266      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5267    AOT_hence ‹∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)› using 2 "&I" by blast
5268    AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5269      by (rule "∃I"(1)) "cqt:2[lambda]"
5270  }
5271  moreover {
5272    AOT_assume 2: ‹¬□∀x([F]x ≡ [λz [F]z & p1]x)›
5273    AOT_hence ‹◇¬∀x([F]x ≡ [λz [F]z & p1]x)›
5274      using "KBasic:11"[THEN "≡E"(1)] by blast
5275    AOT_hence ‹∀x ([F]x ≡ [λz [F]z & p1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p1]x)›
5276      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(1)]] "&I" by blast
5277    AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5278      by (rule "∃I"(1)) "cqt:2[lambda]"
5279  }
5280  ultimately AOT_show ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5281    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5282qed
5283
5284AOT_theorem "eqnotnec:5": ‹∀F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5285proof(rule GEN)
5286  fix F
5287
5288  AOT_have Aux_A: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5289  proof(rule "RM◇"; rule "→I"; rule GEN)
5290    AOT_modally_strict {
5291    fix x
5292    AOT_assume 0: ‹ψ›
5293    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5294      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5295    also AOT_have ‹... ≡ [F]x›
5296      apply (rule "≡I"; rule "→I")
5297      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5298      using 0 "&I" by blast
5299    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5300      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5301    }
5302  qed
5303
5304  AOT_have Aux_B: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5305  proof (rule "RM◇"; rule "→I"; rule GEN)
5306    AOT_modally_strict {
5307      fix x
5308      AOT_assume 0: ‹ψ›
5309      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5310        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5311      also AOT_have ‹... ≡ [F]x›
5312        apply (rule "≡I"; rule "→I")
5313        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5314        apply (rule "∨I"(1)) using 0 "&I" by blast
5315      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5316        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5317    }
5318  qed
5319
5320  AOT_have Aux_C: ‹⊢□ ¬ψ → ¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5321  proof(rule "→I"; rule "raa-cor:2")
5322  AOT_modally_strict {
5323      AOT_assume 0: ‹¬ψ›
5324      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5325      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5326      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5327          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5328      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5329        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5330      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5331        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5332      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5333      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5334      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5335    }
5336  qed
5337
5338  AOT_have Aux_D: ‹∀z ([F]z ≡ [λz [F]z & ψ]z) → (¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5339  proof (rule "→I"; rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5340    AOT_modally_strict {
5341      AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5342      AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5343      AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5344      AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5345      AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5346      AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5347    }
5348  next
5349    AOT_modally_strict {
5350      AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5351      AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5352      AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5353      AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5354      AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5355      AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5356    }
5357  qed
5358
5359  AOT_obtain p1 where p1_prop: ‹¬p1 & ◇p1› using "cont-tf-thm:2" "∃E"[rotated] "cont-tf:2"[THEN "≡dfE"] by blast
5360  {
5361    AOT_assume 1: ‹∀x([F]x ≡ [λz [F]z & p1]x)›
5362    AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5363      using Aux_B[THEN "→E", OF p1_prop[THEN "&E"(2)]].
5364    AOT_have ‹¬∀x([λz [F]z & p1]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5365      using Aux_C[THEN "→E", OF p1_prop[THEN "&E"(1)]].
5366    AOT_hence 3: ‹¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)›
5367      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5368    AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x) & ◇∀x([F]x ≡ [λz [F]z & p1 ∨ ¬p1]x)› using 2 "&I" by blast
5369    AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5370      by (rule "∃I"(1)) "cqt:2[lambda]"
5371  }
5372  moreover {
5373    AOT_assume 2: ‹¬∀x([F]x ≡ [λz [F]z & p1]x)›
5374    AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p1]x)›
5375      using "KBasic:11"[THEN "≡E"(1)] by blast
5376    AOT_hence ‹¬∀x ([F]x ≡ [λz [F]z & p1]x) & ◇∀x([F]x ≡ [λz [F]z & p1]x)›
5377      using Aux_A[THEN "→E", OF p1_prop[THEN "&E"(2)]] "&I" by blast
5378    AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5379      by (rule "∃I"(1)) "cqt:2[lambda]"
5380  }
5381  ultimately AOT_show ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5382    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5383qed
5384
5385AOT_theorem "eqnotnec:6": ‹∀F∃G(𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5386proof(rule GEN)
5387  fix F
5388
5389  AOT_have Aux_A: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5390  proof(rule "RM◇"; rule "→I"; rule GEN)
5391    AOT_modally_strict {
5392    fix x
5393    AOT_assume 0: ‹ψ›
5394    AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5395      by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5396    also AOT_have ‹... ≡ [F]x›
5397      apply (rule "≡I"; rule "→I")
5398      using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5399      using 0 "&I" by blast
5400    finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5401      using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5402    }
5403  qed
5404
5405  AOT_have Aux_B: ‹⊢□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5406  proof (rule "RM◇"; rule "→I"; rule GEN)
5407    AOT_modally_strict {
5408      fix x
5409      AOT_assume 0: ‹ψ›
5410      AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5411        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5412      also AOT_have ‹... ≡ [F]x›
5413        apply (rule "≡I"; rule "→I")
5414        using  "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5415        apply (rule "∨I"(1)) using 0 "&I" by blast
5416      finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5417        using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5418    }
5419  qed
5420
5421  AOT_have Aux_C: ‹⊢□ 𝒜¬ψ → 𝒜¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5422  proof(rule "act-cond"[THEN "→E"]; rule "RA[2]"; rule "→I"; rule "raa-cor:2")
5423  AOT_modally_strict {
5424      AOT_assume 0: ‹¬ψ›
5425      AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5426      AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5427      moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5428          by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5429      moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5430        by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5431      ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5432        using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5433      moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5434      ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5435      AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5436    }
5437  qed
5438
5439  AOT_have Aux_D: ‹𝒜∀z ([F]z ≡ [λz [F]z & ψ]z) → (𝒜¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ 𝒜¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5440  proof (rule "→I"; rule "Act-Basic:5"[THEN "≡E"(1)])
5441    AOT_assume ‹𝒜∀z ([F]z ≡ [λz [F]z & ψ]z)›
5442    AOT_thus ‹𝒜(¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))›
5443    proof (rule "RA[3]"[where Γ="{«∀z ([F]z ≡ [λz [F]z & ψ]z)»}", simplified, rotated])
5444      AOT_modally_strict {
5445        AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5446        AOT_thus ‹¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5447          apply -
5448        proof(rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5449        AOT_modally_strict {
5450          AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5451          AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5452          AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5453          AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5454          AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5455          AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5456        }
5457      next
5458        AOT_modally_strict {
5459          AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5460          AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5461          AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5462          AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5463          AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5464          AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5465        }
5466      qed
5467      }
5468    qed
5469  qed
5470
5471  AOT_have ‹¬𝒜q0›
5472    apply (rule "=dfI"(2)[OF q0_def])
5473     apply (fact "log-prop-prop:2")
5474    by (fact AOT)
5475  AOT_hence q0_prop_1: ‹𝒜¬q0›
5476    using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5477  {
5478    AOT_assume 1: ‹𝒜∀x([F]x ≡ [λz [F]z & q0]x)›
5479    AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5480      using Aux_B[THEN "→E", OF q0_prop[THEN "&E"(1)]].
5481    AOT_have ‹𝒜¬∀x([λz [F]z & q0]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5482      using Aux_C[THEN "→E", OF q0_prop_1].
5483    AOT_hence 3: ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)›
5484      using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5485    AOT_hence ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x) & ◇∀x([F]x ≡ [λz [F]z & q0 ∨ ¬q0]x)› using 2 "&I" by blast
5486    AOT_hence ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5487      by (rule "∃I"(1)) "cqt:2[lambda]"
5488  }
5489  moreover {
5490    AOT_assume 2: ‹¬𝒜∀x([F]x ≡ [λz [F]z & q0]x)›
5491    AOT_hence ‹𝒜¬∀x([F]x ≡ [λz [F]z & q0]x)›
5492      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5493    AOT_hence ‹𝒜¬∀x ([F]x ≡ [λz [F]z & q0]x) & ◇∀x([F]x ≡ [λz [F]z & q0]x)›
5494      using Aux_A[THEN "→E", OF q0_prop[THEN "&E"(1)]] "&I" by blast
5495    AOT_hence ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5496      by (rule "∃I"(1)) "cqt:2[lambda]"
5497  }
5498  ultimately AOT_show ‹∃G (𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5499    using "∨E"(1)[OF "exc-mid"] "→I" by blast
5500qed
5501
5502AOT_theorem "oa-contingent:1": ‹O! ≠ A!›
5503proof(rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
5504  fix x
5505  AOT_assume 1: ‹O! = A!›
5506  AOT_hence ‹[λx ◇E!x] = A!›
5507    by (rule "=dfE"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5508  AOT_hence ‹[λx ◇E!x] = [λx ¬◇E!x]›
5509    by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5510  moreover AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
5511    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5512  ultimately AOT_have ‹[λx ¬◇E!x]x ≡ ◇E!x›
5513    using "rule=E" by fast
5514  moreover AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
5515    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5516  ultimately AOT_have ‹◇E!x ≡ ¬◇E!x› using "≡E"(6) "Commutativity of ≡"[THEN "≡E"(1)] by blast
5517  AOT_thus "(◇E!x ≡ ¬◇E!x) & ¬(◇E!x ≡ ¬◇E!x)" using "oth-class-taut:3:c" "&I" by blast
5518qed
5519
5520AOT_theorem "oa-contingent:2": ‹O!x ≡ ¬A!x›
5521proof -
5522  AOT_have ‹O!x ≡ [λx ◇E!x]x›
5523    apply (rule "≡I"; rule "→I")
5524     apply (rule "=dfE"(2)[OF AOT_ordinary])
5525      apply "cqt:2[lambda]"
5526     apply argo
5527    apply (rule  "=dfI"(2)[OF AOT_ordinary])
5528     apply "cqt:2[lambda]"
5529    by argo
5530  also AOT_have ‹… ≡ ◇E!x›
5531    by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5532  also AOT_have ‹… ≡ ¬¬◇E!x›
5533    using "oth-class-taut:3:b".
5534  also AOT_have ‹… ≡ ¬[λx ¬◇E!x]x›
5535    by (rule "beta-C-meta"[THEN "→E", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric]) "cqt:2[lambda]"
5536  also AOT_have ‹… ≡ ¬A!x›
5537    apply (rule "≡I"; rule "→I")
5538     apply (rule "=dfI"(2)[OF AOT_abstract])
5539      apply "cqt:2[lambda]"
5540     apply argo
5541    apply (rule "=dfE"(2)[OF AOT_abstract])
5542     apply "cqt:2[lambda]"
5543    by argo
5544  finally show ?thesis.
5545qed
5546
5547AOT_theorem "oa-contingent:3": ‹A!x ≡ ¬O!x›
5548  by (AOT_subst ‹A!x› ‹¬¬A!x›)
5549     (auto simp add: "oth-class-taut:3:b" "oa-contingent:2"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric])
5550
5551AOT_theorem "oa-contingent:4": ‹Contingent(O!)›
5552proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:1", THEN "≡E"(2)]; rule "&I")
5553  AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5554  AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5555  then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5556  AOT_hence ‹[λx ◇E!x]a›
5557    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5558  AOT_hence ‹O!a›
5559    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5560  AOT_hence ‹∃x O!x› using "∃I" by blast
5561  AOT_thus ‹◇∃x O!x› using "T◇"[THEN "→E"] by blast
5562next
5563  AOT_obtain a where ‹A!a›
5564    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5565  AOT_hence ‹¬O!a› using "oa-contingent:3"[THEN "≡E"(1)] by blast
5566  AOT_hence ‹∃x ¬O!x› using "∃I" by fast
5567  AOT_thus ‹◇∃x ¬O!x› using "T◇"[THEN "→E"] by blast
5568qed
5569
5570AOT_theorem "oa-contingent:5": ‹Contingent(A!)›
5571proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:2", THEN "≡E"(2)]; rule "&I")
5572  AOT_obtain a where ‹A!a›
5573    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5574  AOT_hence ‹∃x A!x› using "∃I" by fast
5575  AOT_thus ‹◇∃x A!x› using "T◇"[THEN "→E"] by blast
5576next
5577  AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5578  AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5579  then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5580  AOT_hence ‹[λx ◇E!x]a›
5581    by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5582  AOT_hence ‹O!a›
5583    by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5584  AOT_hence ‹¬A!a› using "oa-contingent:2"[THEN "≡E"(1)] by blast
5585  AOT_hence ‹∃x ¬A!x› using "∃I" by fast
5586  AOT_thus ‹◇∃x ¬A!x› using "T◇"[THEN "→E"] by blast
5587qed
5588
5589AOT_theorem "oa-contingent:7": ‹O!-x ≡ ¬A!-x›
5590proof -
5591  AOT_have ‹O!x ≡ ¬A!x›
5592    using "oa-contingent:2" by blast
5593  also AOT_have ‹… ≡ A!-x›
5594    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:2"].
5595  finally AOT_have 1: ‹O!x ≡ A!-x›.
5596
5597  AOT_have ‹A!x ≡ ¬O!x›
5598    using "oa-contingent:3" by blast
5599  also AOT_have ‹… ≡ O!-x›
5600    using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:1"].
5601  finally AOT_have 2: ‹A!x ≡ O!-x›.
5602
5603  AOT_show ‹O!-x ≡ ¬A!-x›
5604    using 1[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "oa-contingent:3"[of _ x] 2[symmetric]
5605          "≡E"(5) by blast
5606qed
5607
5608AOT_theorem "oa-contingent:6": ‹O!- ≠ A!-›
5609proof (rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5610  AOT_assume 1: ‹O!- = A!-›
5611  fix x
5612  AOT_have ‹A!-x ≡ O!-x›
5613    apply (rule "rule=E"[rotated, OF 1]) by (fact "oth-class-taut:3:a")
5614  AOT_hence ‹A!-x ≡ ¬A!-x›
5615    using "oa-contingent:7" "≡E" by fast
5616  AOT_thus ‹(A!-x ≡ ¬A!-x) & ¬(A!-x ≡ ¬A!-x)› using "oth-class-taut:3:c" "&I" by blast
5617qed
5618
5619AOT_theorem "oa-contingent:8": ‹Contingent(O!-)›
5620  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:1", THEN "≡E"(1), OF "oa-contingent:4"].
5621
5622AOT_theorem "oa-contingent:9": ‹Contingent(A!-)›
5623  using "thm-cont-prop:3"[unvarify F, OF "oa-exist:2", THEN "≡E"(1), OF "oa-contingent:5"].
5624
5625AOT_define WeaklyContingent :: ‹Π ⇒ φ› ("WeaklyContingent'(_')")
5626  "df-cont-nec": "WeaklyContingent([F]) ≡df Contingent([F]) & ∀x (◇[F]x → □[F]x)"
5627
5628AOT_theorem "cont-nec-fact1:1": ‹WeaklyContingent([F]) ≡ WeaklyContingent([F]-)›
5629proof -
5630  AOT_have ‹WeaklyContingent([F]) ≡ Contingent([F]) & ∀x (◇[F]x → □[F]x)›
5631    using "df-cont-nec"[THEN "≡Df"] by blast
5632  also AOT_have ‹... ≡ Contingent([F]-) & ∀x (◇[F]x → □[F]x)›
5633    apply (rule "oth-class-taut:8:f"[THEN "≡E"(2)]; rule "→I")
5634    using "thm-cont-prop:3".
5635  also AOT_have ‹… ≡ Contingent([F]-) & ∀x (◇[F]-x → □[F]-x)›
5636  proof (rule "oth-class-taut:8:e"[THEN "≡E"(2)]; rule "→I"; rule "≡I"; rule "→I"; rule GEN; rule "→I")
5637    fix x
5638    AOT_assume 0: ‹∀x (◇[F]x → □[F]x)›
5639    AOT_assume 1: ‹◇[F]-x›
5640    AOT_have ‹◇¬[F]x›
5641      by (AOT_subst (reverse) ‹¬[F]x› ‹[F]-x›)
5642         (auto simp add: "thm-relation-negation:1" 1)
5643    AOT_hence 2: ‹¬□[F]x›
5644      using "KBasic:11"[THEN "≡E"(2)] by blast
5645    AOT_show ‹□[F]-x›
5646    proof (rule "raa-cor:1")
5647      AOT_assume 3: ‹¬□[F]-x›
5648      AOT_have ‹¬□¬[F]x›
5649        by (AOT_subst (reverse) ‹¬[F]x› ‹[F]-x›)
5650           (auto simp add: "thm-relation-negation:1" 3)
5651      AOT_hence ‹◇[F]x›
5652        using "conventions:5"[THEN "≡dfI"] by simp
5653      AOT_hence ‹□[F]x› using 0 "∀E" "→E" by fast
5654      AOT_thus ‹□[F]x & ¬□[F]x› using "&I" 2 by blast
5655    qed
5656  next
5657    fix x
5658    AOT_assume 0: ‹∀x (◇[F]-x → □[F]-x)›
5659    AOT_assume 1: ‹◇[F]x›
5660    AOT_have ‹◇¬[F]-x›
5661      by (AOT_subst ‹¬[F]-x› ‹[F]x›)
5662         (auto simp: "thm-relation-negation:2" 1)
5663    AOT_hence 2: ‹¬□[F]-x›
5664      using "KBasic:11"[THEN "≡E"(2)] by blast
5665    AOT_show ‹□[F]x›
5666    proof (rule "raa-cor:1")
5667      AOT_assume 3: ‹¬□[F]x›
5668      AOT_have ‹¬□¬[F]-x›
5669        by (AOT_subst ‹¬[F]-x› ‹[F]x›)
5670           (auto simp add: "thm-relation-negation:2" 3)
5671      AOT_hence ‹◇[F]-x›
5672        using "conventions:5"[THEN "≡dfI"] by simp
5673      AOT_hence ‹□[F]-x› using 0 "∀E" "→E" by fast
5674      AOT_thus ‹□[F]-x & ¬□[F]-x› using "&I" 2 by blast
5675    qed
5676  qed
5677  also AOT_have ‹… ≡ WeaklyContingent([F]-)›
5678    using "df-cont-nec"[THEN "≡Df", symmetric] by blast
5679  finally show ?thesis.
5680qed
5681
5682AOT_theorem "cont-nec-fact1:2": ‹(WeaklyContingent([F]) & ¬WeaklyContingent([G])) → F ≠ G›
5683proof (rule "→I"; rule "=-infix"[THEN "≡dfI"]; rule "raa-cor:2")
5684  AOT_assume 1: ‹WeaklyContingent([F]) & ¬WeaklyContingent([G])›
5685  AOT_hence ‹WeaklyContingent([F])› using "&E" by blast
5686  moreover AOT_assume ‹F = G›
5687  ultimately AOT_have ‹WeaklyContingent([G])›
5688    using "rule=E" by blast
5689  AOT_thus ‹WeaklyContingent([G]) & ¬WeaklyContingent([G])›
5690    using 1 "&I" "&E" by blast
5691qed
5692
5693AOT_theorem "cont-nec-fact2:1": ‹WeaklyContingent(O!)›
5694proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5695  AOT_show ‹Contingent(O!)›
5696    using "oa-contingent:4".
5697next
5698  AOT_show ‹∀x (◇[O!]x → □[O!]x)›
5699    apply (rule GEN; rule "→I")
5700    using "oa-facts:5"[THEN "≡E"(1)] by blast
5701qed
5702
5703
5704AOT_theorem "cont-nec-fact2:2": ‹WeaklyContingent(A!)›
5705proof (rule "df-cont-nec"[THEN "≡dfI"]; rule "&I")
5706  AOT_show ‹Contingent(A!)›
5707    using "oa-contingent:5".
5708next
5709  AOT_show ‹∀x (◇[A!]x → □[A!]x)›
5710    apply (rule GEN; rule "→I")
5711    using "oa-facts:6"[THEN "≡E"(1)] by blast
5712qed
5713
5714AOT_theorem "cont-nec-fact2:3": ‹¬WeaklyContingent(E!)›
5715proof (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5716       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "raa-cor:2")
5717  AOT_have ‹◇∃x (E!x & ¬𝒜E!x)› using "qml:4"[axiom_inst].
5718  AOT_hence ‹∃x ◇(E!x & ¬𝒜E!x)› using "BF◇"[THEN "→E"] by blast
5719  then AOT_obtain a where ‹◇(E!a & ¬𝒜E!a)› using "∃E"[rotated] by blast
5720  AOT_hence 1: ‹◇E!a & ◇¬𝒜E!a› using "KBasic2:3"[THEN "→E"] by simp
5721  moreover AOT_assume ‹∀x (◇[E!]x → □[E!]x)›
5722  ultimately AOT_have ‹□E!a› using "&E" "∀E" "→E" by fast
5723  AOT_hence ‹𝒜E!a› using "nec-imp-act"[THEN "→E"] by blast
5724  AOT_hence ‹□𝒜E!a› using "qml-act:1"[axiom_inst, THEN "→E"] by blast
5725  moreover AOT_have ‹¬□𝒜E!a› using "KBasic:11"[THEN "≡E"(2)] 1[THEN "&E"(2)] by meson
5726  ultimately AOT_have ‹□𝒜E!a & ¬□𝒜E!a› using "&I" by blast
5727  AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
5728qed
5729
5730AOT_theorem "cont-nec-fact2:4": ‹¬WeaklyContingent(L)›
5731  apply (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5732       rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(1))
5733  apply (rule "contingent-properties:4"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
5734  apply (rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "useful-tautologies:2"[THEN "→E"])
5735  using "thm-noncont-e-e:3"[THEN "contingent-properties:3"[THEN "≡dfE"]].
5736
5737(* TODO: cleanup *)
5738AOT_theorem "cont-nec-fact2:5": ‹O! ≠ E! & O! ≠ E!- & O! ≠ L & O! ≠ L-›
5739proof -
5740  AOT_have 1: ‹L↓›
5741    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5742  {
5743    fix φ and Π Π' :: ‹<κ>›
5744    AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if  ‹φ{Π}› and ‹¬φ{Π'}›
5745    proof (rule "raa-cor:2")
5746      AOT_assume ‹φ{Π'} ≡ φ{Π}›
5747      AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5748      AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5749    qed
5750    AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5751      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5752  } note 0 = this
5753  show ?thesis
5754    apply(safe intro!: "&I"; rule 0)
5755    using "cqt:2[concrete]"[axiom_inst] apply blast
5756    using "oa-exist:1" apply blast
5757    using "cont-nec-fact2:3" apply fast
5758    apply (rule "useful-tautologies:2"[THEN "→E"])
5759    using "cont-nec-fact2:1" apply fast
5760    using "rel-neg-T:3" apply fast
5761    using "oa-exist:1" apply blast
5762    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5763    apply (rule "useful-tautologies:2"[THEN "→E"])
5764    using "cont-nec-fact2:1" apply blast
5765    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5766    using "oa-exist:1" apply fast
5767    using "cont-nec-fact2:4" apply fast
5768    apply (rule "useful-tautologies:2"[THEN "→E"])
5769    using "cont-nec-fact2:1" apply fast
5770    using "rel-neg-T:3" apply fast
5771    using "oa-exist:1" apply fast
5772    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5773    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5774    apply (rule "useful-tautologies:2"[THEN "→E"])
5775    using "cont-nec-fact2:1" by blast
5776qed
5777
5778(* TODO: cleanup together with above *)
5779AOT_theorem "cont-nec-fact2:6": ‹A! ≠ E! & A! ≠ E!- & A! ≠ L & A! ≠ L-›
5780proof -
5781  AOT_have 1: ‹L↓›
5782    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
5783  {
5784    fix φ and Π Π' :: ‹<κ>›
5785    AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if  ‹φ{Π}› and ‹¬φ{Π'}›
5786    proof (rule "raa-cor:2")
5787      AOT_assume ‹φ{Π'} ≡ φ{Π}›
5788      AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5789      AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5790    qed
5791    AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5792      using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5793  } note 0 = this
5794  show ?thesis
5795    apply(safe intro!: "&I"; rule 0)
5796    using "cqt:2[concrete]"[axiom_inst] apply blast
5797    using "oa-exist:2" apply blast
5798    using "cont-nec-fact2:3" apply fast
5799    apply (rule "useful-tautologies:2"[THEN "→E"])
5800    using "cont-nec-fact2:2" apply fast
5801    using "rel-neg-T:3" apply fast
5802    using "oa-exist:2" apply blast
5803    using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5804    apply (rule "useful-tautologies:2"[THEN "→E"])
5805    using "cont-nec-fact2:2" apply blast
5806    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5807    using "oa-exist:2" apply fast
5808    using "cont-nec-fact2:4" apply fast
5809    apply (rule "useful-tautologies:2"[THEN "→E"])
5810    using "cont-nec-fact2:2" apply fast
5811    using "rel-neg-T:3" apply fast
5812    using "oa-exist:2" apply fast
5813    apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5814    apply (rule "=dfI"(2)[OF L_def]; "cqt:2[lambda]")
5815    apply (rule "useful-tautologies:2"[THEN "→E"])
5816    using "cont-nec-fact2:2" by blast
5817qed
5818
5819AOT_define necessary_or_contingently_false :: ‹φ ⇒ φ› ("Δ_" [49] 54)
5820  ‹Δp ≡df □p ∨ (¬𝒜p & ◇p)›
5821
5822AOT_theorem sixteen:
5823 shows ‹∃F1∃F2∃F3∃F4∃F5∃F6∃F7∃F8∃F9∃F10∃F11∃F12∃F13∃F14∃F15∃F16 (
5824«F1::<κ>» ≠ F2 & F1 ≠ F3 & F1 ≠ F4 & F1 ≠ F5 & F1 ≠ F6 & F1 ≠ F7 & F1 ≠ F8 & F1 ≠ F9 & F1 ≠ F10 & F1 ≠ F11 & F1 ≠ F12 & F1 ≠ F13 & F1 ≠ F14 & F1 ≠ F15 & F1 ≠ F16 &
5825F2 ≠ F3 & F2 ≠ F4 & F2 ≠ F5 & F2 ≠ F6 & F2 ≠ F7 & F2 ≠ F8 & F2 ≠ F9 & F2 ≠ F10 & F2 ≠ F11 & F2 ≠ F12 & F2 ≠ F13 & F2 ≠ F14 & F2 ≠ F15 & F2 ≠ F16 &
5826F3 ≠ F4 & F3 ≠ F5 & F3 ≠ F6 & F3 ≠ F7 & F3 ≠ F8 & F3 ≠ F9 & F3 ≠ F10 & F3 ≠ F11 & F3 ≠ F12 & F3 ≠ F13 & F3 ≠ F14 & F3 ≠ F15 & F3 ≠ F16 &
5827F4 ≠ F5 & F4 ≠ F6 & F4 ≠ F7 & F4 ≠ F8 & F4 ≠ F9 & F4 ≠ F10 & F4 ≠ F11 & F4 ≠ F12 & F4 ≠ F13 & F4 ≠ F14 & F4 ≠ F15 & F4 ≠ F16 &
5828F5 ≠ F6 & F5 ≠ F7 & F5 ≠ F8 & F5 ≠ F9 & F5 ≠ F10 & F5 ≠ F11 & F5 ≠ F12 & F5 ≠ F13 & F5 ≠ F14 & F5 ≠ F15 & F5 ≠ F16 &
5829F6 ≠ F7 & F6 ≠ F8 & F6 ≠ F9 & F6 ≠ F10 & F6 ≠ F11 & F6 ≠ F12 & F6 ≠ F13 & F6 ≠ F14 & F6 ≠ F15 & F6 ≠ F16 &
5830F7 ≠ F8 & F7 ≠ F9 & F7 ≠ F10 & F7 ≠ F11 & F7 ≠ F12 & F7 ≠ F13 & F7 ≠ F14 & F7 ≠ F15 & F7 ≠ F16 &
5831F8 ≠ F9 & F8 ≠ F10 & F8 ≠ F11 & F8 ≠ F12 & F8 ≠ F13 & F8 ≠ F14 & F8 ≠ F15 & F8 ≠ F16 &
5832F9 ≠ F10 & F9 ≠ F11 & F9 ≠ F12 & F9 ≠ F13 & F9 ≠ F14 & F9 ≠ F15 & F9 ≠ F16 &
5833F10 ≠ F11 & F10 ≠ F12 & F10 ≠ F13 & F10 ≠ F14 & F10 ≠ F15 & F10 ≠ F16 &
5834F11 ≠ F12 & F11 ≠ F13 & F11 ≠ F14 & F11 ≠ F15 & F11 ≠ F16 &
5835F12 ≠ F13 & F12 ≠ F14 & F12 ≠ F15 & F12 ≠ F16 &
5836F13 ≠ F14 & F13 ≠ F15 & F13 ≠ F16 &
5837F14 ≠ F15 & F14 ≠ F16 &
5838F15 ≠ F16)› 
5839proof -
5840
5841  AOT_have Delta_pos: ‹Δφ → ◇φ› for φ
5842  proof(rule "→I")
5843    AOT_assume ‹Δφ›
5844    AOT_hence ‹□φ ∨ (¬𝒜φ & ◇φ)›
5845      using "≡dfE"[OF necessary_or_contingently_false] by blast
5846    moreover {
5847      AOT_assume ‹□φ›
5848      AOT_hence ‹◇φ›
5849        by (metis "B◇" "T◇" "vdash-properties:10")
5850    }
5851    moreover {
5852      AOT_assume ‹¬𝒜φ & ◇φ›
5853      AOT_hence ‹◇φ›
5854        using "&E" by blast
5855    }
5856    ultimately AOT_show ‹◇φ›
5857      by (metis "∨E"(2) "raa-cor:1") 
5858  qed
5859
5860  AOT_have act_and_not_nec_not_delta: ‹¬Δφ› if ‹𝒜φ› and ‹¬□φ› for φ
5861    using "≡dfE" "&E"(1) "∨E"(2) necessary_or_contingently_false "raa-cor:3" that(1) that(2) by blast
5862  AOT_have act_and_pos_not_not_delta: ‹¬Δφ› if ‹𝒜φ› and ‹◇¬φ› for φ
5863    using "KBasic:11" act_and_not_nec_not_delta "≡E"(2) that(1) that(2) by blast
5864  AOT_have impossible_delta: ‹¬Δφ› if ‹¬◇φ› for φ
5865    using Delta_pos "modus-tollens:1" that by blast
5866  AOT_have not_act_and_pos_delta: ‹Δφ› if ‹¬𝒜φ› and ‹◇φ› for φ
5867    by (meson "≡dfI" "&I" "∨I"(2) necessary_or_contingently_false that(1) that(2))
5868  AOT_have nec_delta: ‹Δφ› if ‹□φ› for φ
5869    using "≡dfI" "∨I"(1) necessary_or_contingently_false that by blast
5870
5871  AOT_obtain a where a_prop: ‹A!a›
5872    using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5873  AOT_obtain b where b_prop: ‹◇[E!]b & ¬𝒜[E!]b›
5874    using "pos-not-pna:3" using "∃E"[rotated] by blast
5875
5876  AOT_have b_ord: ‹[O!]b›
5877  proof(rule "=dfI"(2)[OF AOT_ordinary])
5878    AOT_show ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
5879  next
5880    AOT_show ‹[λx ◇[E!]x]b›
5881    proof (rule "β←C"(1); ("cqt:2[lambda]")?)
5882      AOT_show ‹b↓› by (rule "cqt:2[const_var]"[axiom_inst])
5883      AOT_show ‹◇[E!]b› by (fact b_prop[THEN "&E"(1)])
5884    qed
5885  qed
5886
5887  AOT_have nec_not_L_neg: ‹□¬[L-]x› for x
5888    using "thm-noncont-e-e:2" "contingent-properties:2"[THEN "≡dfE"] "&E"
5889          CBF[THEN "→E"] "∀E" by blast
5890  AOT_have nec_L: ‹□[L]x› for x
5891    using "thm-noncont-e-e:1" "contingent-properties:1"[THEN "≡dfE"]
5892      CBF[THEN "→E"] "∀E" by blast
5893
5894  AOT_have act_ord_b: ‹𝒜[O!]b›
5895    using b_ord "≡E"(1) "oa-facts:7" by blast
5896  AOT_have delta_ord_b: ‹Δ[O!]b›
5897    by (meson "≡dfI" b_ord "∨I"(1) necessary_or_contingently_false "oa-facts:1" "vdash-properties:10")
5898  AOT_have not_act_ord_a: ‹¬𝒜[O!]a›
5899    by (meson a_prop "≡E"(1) "≡E"(3) "oa-contingent:3" "oa-facts:7")
5900  AOT_have not_delta_ord_a: ‹¬Δ[O!]a›
5901    by (metis Delta_pos "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "reductio-aa:1" "vdash-properties:10")
5902
5903  AOT_have not_act_abs_b: ‹¬𝒜[A!]b›
5904    by (meson b_ord "≡E"(1) "≡E"(3) "oa-contingent:2" "oa-facts:8")
5905  AOT_have not_delta_abs_b: ‹¬Δ[A!]b›
5906  proof(rule "raa-cor:2")
5907    AOT_assume ‹Δ[A!]b›
5908    AOT_hence ‹◇[A!]b›
5909      by (metis Delta_pos "vdash-properties:10")
5910    AOT_thus ‹[A!]b & ¬[A!]b›
5911      by (metis b_ord "&I" "≡E"(1) "oa-contingent:2" "oa-facts:4" "vdash-properties:10")
5912  qed
5913  AOT_have act_abs_a: ‹𝒜[A!]a›
5914    using a_prop "≡E"(1) "oa-facts:8" by blast
5915  AOT_have delta_abs_a: ‹Δ[A!]a›
5916      by (metis "≡dfI" a_prop "oa-facts:2" "vdash-properties:10" "∨I"(1) necessary_or_contingently_false)
5917
5918  AOT_have not_act_concrete_b: ‹¬𝒜[E!]b›
5919    using b_prop "&E"(2) by blast
5920  AOT_have delta_concrete_b: ‹Δ[E!]b›
5921  proof (rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5922    AOT_show ‹¬𝒜[E!]b› using b_prop "&E"(2) by blast
5923  next
5924    AOT_show ‹◇[E!]b› using b_prop "&E"(1) by blast
5925  qed
5926  AOT_have not_act_concrete_a: ‹¬𝒜[E!]a›
5927  proof (rule "raa-cor:2")
5928    AOT_assume ‹𝒜[E!]a›
5929    AOT_hence 1: ‹◇[E!]a› by (metis "Act-Sub:3" "vdash-properties:10")
5930    AOT_have ‹[A!]a› by (simp add: a_prop)
5931    AOT_hence ‹[λx ¬◇[E!]x]a›
5932      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5933    AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5934    AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5935  qed
5936  AOT_have not_delta_concrete_a: ‹¬Δ[E!]a›
5937  proof (rule "raa-cor:2")
5938    AOT_assume ‹Δ[E!]a›
5939    AOT_hence 1: ‹◇[E!]a› by (metis Delta_pos "vdash-properties:10")
5940    AOT_have ‹[A!]a› by (simp add: a_prop)
5941    AOT_hence ‹[λx ¬◇[E!]x]a›
5942      by (rule "=dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5943    AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5944    AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5945  qed
5946
5947  AOT_have not_act_q_zero: ‹¬𝒜q0›
5948    by (meson "log-prop-prop:2" "pos-not-pna:1" q0_def "reductio-aa:1" "rule-id-df:2:a[zero]")
5949  AOT_have delta_q_zero: ‹Δq0›
5950  proof(rule "≡dfI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5951    AOT_show ‹¬𝒜q0› using not_act_q_zero.
5952    AOT_show ‹◇q0› by (meson "&E"(1) q0_prop)
5953  qed
5954  AOT_have act_not_q_zero: ‹𝒜¬q0› using "Act-Basic:1" "∨E"(2) not_act_q_zero by blast
5955  AOT_have not_delta_not_q_zero: ‹¬Δ¬q0›
5956      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta "&E"(1) "∨E"(2) not_act_q_zero q0_prop by blast
5957
5958  AOT_have ‹[L-]↓› by (simp add: "rel-neg-T:3")
5959  moreover AOT_have ‹¬𝒜[L-]b & ¬Δ[L-]b & ¬𝒜[L-]a & ¬Δ[L-]a›
5960  proof (safe intro!: "&I")
5961    AOT_show ‹¬𝒜[L-]b› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5962    AOT_show ‹¬Δ[L-]b› by (meson Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg)
5963    AOT_show ‹¬𝒜[L-]a› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5964    AOT_show ‹¬Δ[L-]a› using Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg by blast
5965  qed
5966  ultimately AOT_obtain F0 where ‹¬𝒜[F0]b & ¬Δ[F0]b & ¬𝒜[F0]a & ¬Δ[F0]a›
5967    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5968  AOT_hence ‹¬𝒜[F0]b› and ‹¬Δ[F0]b› and ‹¬𝒜[F0]a› and ‹¬Δ[F0]a›
5969    using "&E" by blast+
5970  note props = this
5971
5972  let ?Π = "«[λy [A!]y & q0]»"
5973  AOT_modally_strict {
5974    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5975  } note 1 = this
5976  moreover AOT_have‹¬𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
5977  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5978    AOT_show ‹¬𝒜([A!]b & q0)›
5979      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5980  next AOT_show ‹¬Δ([A!]b & q0)›
5981      by (metis Delta_pos "KBasic2:3" "&E"(1) "≡E"(4) not_act_abs_b "oa-facts:4" "oa-facts:8" "raa-cor:3" "vdash-properties:10")
5982  next AOT_show ‹¬𝒜([A!]a & q0)›
5983      using "Act-Basic:2" "&E"(2) "≡E"(1) not_act_q_zero "raa-cor:3" by blast
5984  next AOT_show ‹Δ([A!]a & q0)›
5985    proof (rule not_act_and_pos_delta)
5986      AOT_show ‹¬𝒜([A!]a & q0)›
5987        using "Act-Basic:2" "&E"(2) "≡E"(4) not_act_q_zero "raa-cor:3" by blast
5988    next AOT_show ‹◇([A!]a & q0)›
5989        by (metis "&I" "→E" Delta_pos "KBasic:16" "&E"(1) delta_abs_a "≡E"(1) "oa-facts:6" q0_prop)
5990    qed
5991  qed
5992  ultimately AOT_obtain F1 where ‹¬𝒜[F1]b & ¬Δ[F1]b & ¬𝒜[F1]a & Δ[F1]a›
5993    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5994  AOT_hence ‹¬𝒜[F1]b› and ‹¬Δ[F1]b› and ‹¬𝒜[F1]a› and ‹Δ[F1]a›
5995    using "&E" by blast+
5996  note props = props this
5997
5998  let ?Π = "«[λy [A!]y & ¬q0]»"
5999  AOT_modally_strict {
6000    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6001  } note 1 = this
6002  moreover AOT_have ‹¬𝒜[«?Π»]b & ¬Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6003  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6004    AOT_show ‹¬𝒜([A!]b & ¬q0)›
6005      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
6006  next AOT_show ‹¬Δ([A!]b & ¬q0)›
6007      by (meson "RM◇" Delta_pos "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b "oa-facts:4" "oa-facts:8")
6008  next AOT_show ‹𝒜([A!]a & ¬q0)›
6009      by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
6010  next AOT_show ‹¬Δ([A!]a & ¬q0)›
6011    proof (rule act_and_not_nec_not_delta)
6012      AOT_show ‹𝒜([A!]a & ¬q0)›
6013        by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
6014    next
6015      AOT_show ‹¬□([A!]a & ¬q0)›
6016        by (metis "KBasic2:1" "KBasic:3" "&E"(1) "&E"(2) "≡E"(4) q0_prop "raa-cor:3")
6017    qed
6018  qed
6019  ultimately AOT_obtain F2 where ‹¬𝒜[F2]b & ¬Δ[F2]b & 𝒜[F2]a & ¬Δ[F2]a›
6020    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6021  AOT_hence ‹¬𝒜[F2]b› and ‹¬Δ[F2]b› and ‹𝒜[F2]a› and ‹¬Δ[F2]a›
6022    using "&E" by blast+
6023  note props = props this
6024
6025  AOT_have abstract_prop: ‹¬𝒜[A!]b & ¬Δ[A!]b & 𝒜[A!]a & Δ[A!]a›
6026    using act_abs_a "&I" delta_abs_a not_act_abs_b not_delta_abs_b by presburger
6027  then AOT_obtain F3 where ‹¬𝒜[F3]b & ¬Δ[F3]b & 𝒜[F3]a & Δ[F3]a›
6028    using "∃I"(1)[rotated, THEN "∃E"[rotated]] "oa-exist:2" by fastforce
6029  AOT_hence ‹¬𝒜[F3]b› and ‹¬Δ[F3]b› and ‹𝒜[F3]a› and ‹Δ[F3]a›
6030    using "&E" by blast+
6031  note props = props this
6032
6033  AOT_have ‹¬𝒜[E!]b & Δ[E!]b & ¬𝒜[E!]a & ¬Δ[E!]a›
6034    by (meson "&I" delta_concrete_b not_act_concrete_a not_act_concrete_b not_delta_concrete_a)
6035  then AOT_obtain F4 where ‹¬𝒜[F4]b & Δ[F4]b & ¬𝒜[F4]a & ¬Δ[F4]a›
6036    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6037  AOT_hence ‹¬𝒜[F4]b› and ‹Δ[F4]b› and ‹¬𝒜[F4]a› and ‹¬Δ[F4]a›
6038    using "&E" by blast+
6039  note props = props this
6040
6041  AOT_modally_strict {
6042    AOT_have ‹[λy q0]↓› by "cqt:2[lambda]"
6043  } note 1 = this
6044  moreover AOT_have ‹¬𝒜[λy q0]b & Δ[λy q0]b & ¬𝒜[λy q0]a & Δ[λy q0]a›
6045    by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6046       (auto simp: not_act_q_zero delta_q_zero)
6047  ultimately AOT_obtain F5 where ‹¬𝒜[F5]b & Δ[F5]b & ¬𝒜[F5]a & Δ[F5]a›
6048    using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6049  AOT_hence ‹¬𝒜[F5]b› and ‹Δ[F5]b› and ‹¬𝒜[F5]a› and ‹Δ[F5]a›
6050    using "&E" by blast+
6051  note props = props this
6052
6053  let ?Π = "«[λy [E!]y ∨ ([A!]y & ¬q0)]»"
6054  AOT_modally_strict {
6055    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6056  } note 1 = this
6057  moreover AOT_have ‹¬𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6058  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6059    AOT_have ‹𝒜¬([A!]b & ¬q0)›
6060      by (metis "Act-Basic:1" "Act-Basic:2" abstract_prop "&E"(1) "∨E"(2)
6061                "≡E"(1) "raa-cor:3")
6062    moreover AOT_have ‹¬𝒜[E!]b›
6063      using b_prop "&E"(2) by blast
6064    ultimately AOT_have 2: ‹𝒜(¬[E!]b & ¬([A!]b & ¬q0))›
6065      by (metis "Act-Basic:2" "Act-Sub:1" "&I" "≡E"(3) "raa-cor:1")
6066    AOT_have ‹𝒜¬([E!]b ∨ ([A!]b & ¬q0))›
6067      by (AOT_subst ‹¬([E!]b ∨ ([A!]b & ¬q0))› ‹¬[E!]b & ¬([A!]b & ¬q0)›)
6068         (auto simp: "oth-class-taut:5:d" 2)
6069    AOT_thus ‹¬𝒜([E!]b ∨ ([A!]b & ¬q0))›
6070      by (metis "¬¬I" "Act-Sub:1" "≡E"(4))
6071  next
6072    AOT_show ‹Δ([E!]b ∨ ([A!]b & ¬q0))›
6073    proof (rule not_act_and_pos_delta)
6074      AOT_show ‹¬𝒜([E!]b ∨ ([A!]b & ¬q0))›
6075        by (metis "Act-Basic:2" "Act-Basic:9" "∨E"(2) "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b not_act_concrete_b "raa-cor:3")
6076    next
6077      AOT_show ‹◇([E!]b ∨ ([A!]b & ¬q0))›
6078        using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6079    qed
6080  next AOT_show ‹𝒜([E!]a ∨ ([A!]a & ¬q0))›
6081      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6082  next AOT_show ‹¬Δ([E!]a ∨ ([A!]a & ¬q0))›
6083    proof (rule act_and_not_nec_not_delta)
6084      AOT_show ‹𝒜([E!]a ∨ ([A!]a & ¬q0))›
6085        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6086    next
6087      AOT_have ‹□¬[E!]a›
6088        by (metis "≡dfI" "conventions:5" "&I" "∨I"(2) necessary_or_contingently_false not_act_concrete_a not_delta_concrete_a "raa-cor:3")
6089      moreover AOT_have ‹◇¬([A!]a & ¬q0)›
6090        by (metis "KBasic2:1" "KBasic:11" "KBasic:3" "&E"(1) "&E"(2) "≡E"(1) q0_prop "raa-cor:3")
6091      ultimately AOT_have ‹◇(¬[E!]a & ¬([A!]a & ¬q0))› by (metis "KBasic:16" "&I" "vdash-properties:10")
6092      AOT_hence ‹◇¬([E!]a ∨ ([A!]a & ¬q0))›
6093        by (metis "RE◇" "≡E"(2) "oth-class-taut:5:d")
6094      AOT_thus ‹¬□([E!]a ∨ ([A!]a & ¬q0))› by (metis "KBasic:12" "≡E"(1) "raa-cor:3")
6095    qed
6096  qed
6097  ultimately AOT_obtain F6 where ‹¬𝒜[F6]b & Δ[F6]b & 𝒜[F6]a & ¬Δ[F6]a›
6098    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6099  AOT_hence ‹¬𝒜[F6]b› and ‹Δ[F6]b› and ‹𝒜[F6]a› and ‹¬Δ[F6]a›
6100    using "&E" by blast+
6101  note props = props this
6102
6103  let ?Π = "«[λy [A!]y ∨ [E!]y]»"
6104  AOT_modally_strict {
6105    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6106  } note 1 = this
6107  moreover AOT_have ‹¬𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & Δ[«?Π»]a›
6108  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6109    AOT_show ‹¬𝒜([A!]b ∨ [E!]b)›
6110      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6111  next AOT_show ‹Δ([A!]b ∨ [E!]b)›
6112    proof (rule not_act_and_pos_delta)
6113      AOT_show ‹¬𝒜([A!]b ∨ [E!]b)›
6114        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6115    next AOT_show ‹◇([A!]b ∨ [E!]b)›
6116        using "KBasic2:2" b_prop "&E"(1) "∨I"(2) "≡E"(2) by blast
6117    qed
6118  next AOT_show ‹𝒜([A!]a ∨ [E!]a)›
6119      by (meson "Act-Basic:9" act_abs_a "∨I"(1) "≡E"(2))
6120  next AOT_show ‹Δ([A!]a ∨ [E!]a) ›
6121    proof (rule nec_delta)
6122      AOT_show ‹□([A!]a ∨ [E!]a)›
6123        by (metis "KBasic:15" act_abs_a act_and_not_nec_not_delta "Disjunction Addition"(1) delta_abs_a "raa-cor:3" "vdash-properties:10")
6124    qed
6125  qed
6126  ultimately AOT_obtain F7 where ‹¬𝒜[F7]b & Δ[F7]b & 𝒜[F7]a & Δ[F7]a›
6127    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6128  AOT_hence ‹¬𝒜[F7]b› and ‹Δ[F7]b› and ‹𝒜[F7]a› and ‹Δ[F7]a›
6129    using "&E" by blast+
6130  note props = props this
6131
6132  let ?Π = "«[λy [O!]y & ¬[E!]y]»"
6133  AOT_modally_strict {
6134    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6135  } note 1 = this
6136  moreover AOT_have ‹𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6137  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6138    AOT_show ‹𝒜([O!]b & ¬[E!]b)›
6139      by (metis "Act-Basic:1" "Act-Basic:2" act_ord_b "&I" "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:3")
6140  next AOT_show ‹¬Δ([O!]b & ¬[E!]b)›
6141      by (metis (no_types, hide_lams) "conventions:5" "Act-Sub:1" "RM:1" act_and_not_nec_not_delta "act-conj-act:3"
6142                act_ord_b b_prop "&I" "&E"(1) "Conjunction Simplification"(2) "df-rules-formulas[3]"
6143                "≡E"(3) "raa-cor:1" "→E")
6144  next AOT_show ‹¬𝒜([O!]a & ¬[E!]a)›
6145      using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_ord_a "raa-cor:3" by blast
6146  next AOT_have ‹¬◇([O!]a & ¬[E!]a)›
6147      by (metis "KBasic2:3" "&E"(1) "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "raa-cor:3" "vdash-properties:10")
6148    AOT_thus ‹¬Δ([O!]a & ¬[E!]a)›
6149      by (rule impossible_delta)
6150  qed      
6151  ultimately AOT_obtain F8 where ‹𝒜[F8]b & ¬Δ[F8]b & ¬𝒜[F8]a & ¬Δ[F8]a›
6152    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6153  AOT_hence ‹𝒜[F8]b› and ‹¬Δ[F8]b› and ‹¬𝒜[F8]a› and ‹¬Δ[F8]a›
6154    using "&E" by blast+
6155  note props = props this
6156
6157  (* TODO_PLM: binary property 9 wrong in PLM *)
6158  let ?Π = "«[λy ¬[E!]y & ([O!]y ∨ q0)]»"
6159  AOT_modally_strict {
6160    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6161  } note 1 = this
6162  moreover AOT_have ‹𝒜[«?Π»]b & ¬Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
6163  proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6164    AOT_show ‹𝒜(¬[E!]b & ([O!]b ∨ q0))›
6165      by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6166                "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6167  next AOT_show ‹¬Δ(¬[E!]b & ([O!]b ∨ q0))›
6168    proof (rule act_and_pos_not_not_delta)
6169      AOT_show ‹𝒜(¬[E!]b & ([O!]b ∨ q0))›
6170        by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6171                  "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6172    next
6173      AOT_show ‹◇¬(¬[E!]b & ([O!]b ∨ q0))›
6174      proof (AOT_subst ‹¬(¬[E!]b & ([O!]b ∨ q0))› ‹[E!]b ∨ ¬([O!]b ∨ q0)›)
6175        AOT_modally_strict {
6176          AOT_show ‹¬(¬[E!]b & ([O!]b ∨ q0)) ≡ [E!]b ∨ ¬([O!]b ∨ q0)›
6177            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2) "∨E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
6178        }
6179      next
6180        AOT_show ‹◇([E!]b ∨ ¬([O!]b ∨ q0))›
6181          using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6182       qed
6183     qed
6184   next
6185     AOT_show ‹¬𝒜(¬[E!]a & ([O!]a ∨ q0))›
6186       using "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2" by blast
6187   next
6188     AOT_show ‹Δ(¬[E!]a & ([O!]a ∨ q0))›
6189     proof (rule not_act_and_pos_delta)
6190       AOT_show ‹¬𝒜(¬[E!]a & ([O!]a ∨ q0))›
6191         by (metis "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2")
6192     next
6193       AOT_have ‹□¬[E!]a›
6194         using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "raa-cor:5" by blast
6195       moreover AOT_have ‹◇([O!]a ∨ q0)›
6196         by (metis "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(3) q0_prop "raa-cor:3")
6197       ultimately AOT_show ‹◇(¬[E!]a & ([O!]a ∨ q0))›
6198         by (metis "KBasic:16" "&I" "vdash-properties:10")
6199     qed
6200   qed
6201  ultimately AOT_obtain F9 where ‹𝒜[F9]b & ¬Δ[F9]b & ¬𝒜[F9]a & Δ[F9]a›
6202    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6203  AOT_hence ‹𝒜[F9]b› and ‹¬Δ[F9]b› and ‹¬𝒜[F9]a› and ‹Δ[F9]a›
6204    using "&E" by blast+
6205  note props = props this
6206
6207  AOT_modally_strict {
6208    AOT_have ‹[λy ¬q0]↓› by "cqt:2[lambda]"
6209  } note 1 = this
6210  moreover AOT_have ‹𝒜[λy ¬q0]b & ¬Δ[λy ¬q0]b & 𝒜[λy ¬q0]a & ¬Δ[λy ¬q0]a›
6211    by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1]; auto simp: act_not_q_zero not_delta_not_q_zero)
6212  ultimately AOT_obtain F10 where ‹𝒜[F10]b & ¬Δ[F10]b & 𝒜[F10]a & ¬Δ[F10]a›
6213    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6214  AOT_hence ‹𝒜[F10]b› and ‹¬Δ[F10]b› and ‹𝒜[F10]a› and ‹¬Δ[F10]a›
6215    using "&E" by blast+
6216  note props = props this
6217
6218  AOT_modally_strict {
6219    AOT_have ‹[λy ¬[E!]y]↓› by "cqt:2[lambda]"
6220  } note 1 = this
6221  moreover AOT_have ‹𝒜[λy ¬[E!]y]b & ¬Δ[λy ¬[E!]y]b & 𝒜[λy ¬[E!]y]a & Δ[λy ¬[E!]y]a›
6222  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6223    AOT_show ‹𝒜¬[E!]b›
6224      using "Act-Basic:1" "∨E"(2) not_act_concrete_b by blast
6225  next AOT_show ‹¬Δ¬[E!]b›
6226      using "≡dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta b_prop "&E"(1) "∨E"(2) not_act_concrete_b by blast
6227  next AOT_show ‹𝒜¬[E!]a›
6228      using "Act-Basic:1" "∨E"(2) not_act_concrete_a by blast
6229  next AOT_show ‹Δ¬[E!]a›
6230      using "KBasic2:1" "≡E"(2) nec_delta not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "reductio-aa:1" by blast
6231  qed
6232  ultimately AOT_obtain F11 where ‹𝒜[F11]b & ¬Δ[F11]b & 𝒜[F11]a & Δ[F11]a›
6233    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6234  AOT_hence ‹𝒜[F11]b› and ‹¬Δ[F11]b› and ‹𝒜[F11]a› and ‹Δ[F11]a›
6235    using "&E" by blast+
6236  note props = props this
6237
6238  AOT_have ‹𝒜[O!]b & Δ[O!]b & ¬𝒜[O!]a & ¬Δ[O!]a›
6239    by (simp add: act_ord_b "&I" delta_ord_b not_act_ord_a not_delta_ord_a)
6240  then AOT_obtain F12 where ‹𝒜[F12]b & Δ[F12]b & ¬𝒜[F12]a & ¬Δ[F12]a›
6241    using "oa-exist:1" "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6242  AOT_hence ‹𝒜[F12]b› and ‹Δ[F12]b› and ‹¬𝒜[F12]a› and ‹¬Δ[F12]a›
6243    using "&E" by blast+
6244  note props = props this
6245
6246  let ?Π = "«[λy [O!]y ∨ q0]»"
6247  AOT_modally_strict {
6248    AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6249  } note 1 = this
6250  moreover AOT_have ‹𝒜[«?Π»]b & Δ[«?Π»]b & ¬𝒜[«?Π»]a & Δ[«?Π»]a›
6251  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6252    AOT_show ‹𝒜([O!]b ∨ q0)›
6253      by (meson "Act-Basic:9" act_ord_b "∨I"(1) "≡E"(2))
6254  next AOT_show ‹Δ([O!]b ∨ q0)›
6255      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6256  next AOT_show ‹¬𝒜([O!]a ∨ q0)›
6257      using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6258  next AOT_show ‹Δ([O!]a ∨ q0)›
6259    proof (rule not_act_and_pos_delta)
6260      AOT_show ‹¬𝒜([O!]a ∨ q0)›
6261        using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6262    next AOT_show ‹◇([O!]a ∨ q0)›
6263        using "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(2) q0_prop by blast
6264    qed
6265  qed
6266  ultimately AOT_obtain F13 where ‹𝒜[F13]b & Δ[F13]b & ¬𝒜[F13]a & Δ[F13]a›
6267    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6268  AOT_hence ‹𝒜[F13]b› and ‹Δ[F13]b› and ‹¬𝒜[F13]a› and ‹Δ[F13]a›
6269    using "&E" by blast+
6270  note props = props this
6271
6272  let ?Π = "«[λy [O!]y ∨ ¬q0]»"
6273  AOT_modally_strict {
6274     AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6275  } note 1 = this
6276  moreover AOT_have ‹𝒜[«?Π»]b & Δ[«?Π»]b & 𝒜[«?Π»]a & ¬Δ[«?Π»]a›
6277  proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6278    AOT_show ‹𝒜([O!]b ∨ ¬q0)›
6279      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6280  next AOT_show ‹Δ([O!]b ∨ ¬q0)›
6281      by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6282  next AOT_show ‹𝒜([O!]a ∨ ¬q0)›
6283      by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6284  next AOT_show ‹¬Δ([O!]a ∨ ¬q0)›
6285    proof(rule act_and_pos_not_not_delta)
6286      AOT_show ‹𝒜([O!]a ∨ ¬q0)›
6287        by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6288    next
6289      AOT_have ‹□¬[O!]a›
6290        using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_ord_a not_delta_ord_a "raa-cor:6" by blast
6291      moreover AOT_have ‹◇q0›
6292        by (meson "&E"(1) q0_prop)
6293      ultimately AOT_have 2: ‹◇(¬[O!]a & q0)›
6294         by (metis "KBasic:16" "&I" "vdash-properties:10")
6295      AOT_show ‹◇¬([O!]a ∨ ¬q0)›
6296      proof (AOT_subst (reverse) ‹¬([O!]a ∨ ¬q0)› ‹¬[O!]a & q0›)
6297        AOT_modally_strict {
6298          AOT_show ‹¬[O!]a & q0 ≡ ¬([O!]a ∨ ¬q0)›
6299            by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2)
6300                      "∨E"(3) "deduction-theorem" "≡I" "raa-cor:3")
6301        }
6302      next
6303        AOT_show ‹◇(¬[O!]a & q0)›
6304          using "2" by blast
6305      qed
6306    qed
6307  qed
6308  ultimately AOT_obtain F14 where ‹𝒜[F14]b & Δ[F14]b & 𝒜[F14]a & ¬Δ[F14]a›
6309    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6310  AOT_hence ‹𝒜[F14]b› and ‹Δ[F14]b› and ‹𝒜[F14]a› and ‹¬Δ[F14]a›
6311    using "&E" by blast+
6312  note props = props this
6313
6314  AOT_have ‹[L]↓›
6315    by (rule "=dfI"(2)[OF L_def]) "cqt:2[lambda]"+
6316  moreover AOT_have ‹𝒜[L]b & Δ[L]b & 𝒜[L]a & Δ[L]a›
6317  proof (safe intro!: "&I")
6318    AOT_show ‹𝒜[L]b›
6319      by (meson nec_L "nec-imp-act" "vdash-properties:10")
6320    next AOT_show ‹Δ[L]b› using nec_L nec_delta by blast
6321    next AOT_show ‹𝒜[L]a› by (meson nec_L "nec-imp-act" "vdash-properties:10")
6322    next AOT_show ‹Δ[L]a› using nec_L nec_delta by blast
6323  qed
6324  ultimately AOT_obtain F15 where ‹𝒜[F15]b & Δ[F15]b & 𝒜[F15]a & Δ[F15]a›
6325    using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6326  AOT_hence ‹𝒜[F15]b› and ‹Δ[F15]b› and ‹𝒜[F15]a› and ‹Δ[F15]a›
6327    using "&E" by blast+
6328  note props = props this
6329
6330  show ?thesis
6331    by (rule "∃I"(2)[where β=F0]; rule "∃I"(2)[where β=F1]; rule "∃I"(2)[where β=F2];
6332           rule "∃I"(2)[where β=F3]; rule "∃I"(2)[where β=F4]; rule "∃I"(2)[where β=F5];
6333           rule "∃I"(2)[where β=F6]; rule "∃I"(2)[where β=F7]; rule "∃I"(2)[where β=F8];
6334           rule "∃I"(2)[where β=F9]; rule "∃I"(2)[where β=F10]; rule "∃I"(2)[where β=F11];
6335           rule "∃I"(2)[where β=F12]; rule "∃I"(2)[where β=F13]; rule "∃I"(2)[where β=F14];
6336           rule "∃I"(2)[where β=F15]; safe intro!: "&I")
6337       (match conclusion in "[?v ⊨ [F] ≠ [G]]" for F G ⇒ ‹
6338        match props in A: "[?v ⊨ ¬φ{F}]" for φ ⇒ ‹
6339        match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail› ¦ _ ⇒ ‹
6340        match props in B: "[?v ⊨ φ{G}]" ⇒ ‹
6341        fact "pos-not-equiv-ne:4"[where F=F and G=G and φ=φ, THEN "→E",
6342                                OF "oth-class-taut:4:h"[THEN "≡E"(2)],
6343                                OF "Disjunction Addition"(2)[THEN "→E"],
6344                                OF "&I", OF A, OF B]››››)+
6345qed
6346
6347AOT_theorem "o-objects-exist:1": ‹□∃x O!x›
6348proof(rule RN)
6349  AOT_modally_strict {
6350    AOT_obtain a where ‹◇(E!a & ¬𝒜[E!]a)›
6351      using "∃E"[rotated, OF "qml:4"[axiom_inst, THEN "BF◇"[THEN "→E"]]] by blast
6352    AOT_hence 1: ‹◇E!a› by (metis "KBasic2:3" "&E"(1) "→E")
6353    AOT_have ‹[λx ◇[E!]x]a›
6354    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6355      AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst] by blast
6356    next
6357      AOT_show ‹◇E!a› by (fact 1)
6358    qed
6359    AOT_hence ‹O!a› by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6360    AOT_thus ‹∃x [O!]x› by (rule "∃I")
6361  }
6362qed
6363
6364AOT_theorem "o-objects-exist:2": ‹□∃x A!x›
6365proof (rule RN)
6366  AOT_modally_strict {
6367    AOT_obtain a where ‹[A!]a›
6368      using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
6369    AOT_thus ‹∃x A!x› using "∃I" by blast
6370  }
6371qed
6372
6373AOT_theorem "o-objects-exist:3": ‹□¬∀x O!x›
6374  by (rule RN) (metis (no_types, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(4) "modus-tollens:1" "o-objects-exist:2" "oa-contingent:2" "qml:2"[axiom_inst] "reductio-aa:2")
6375
6376AOT_theorem "o-objects-exist:4": ‹□¬∀x A!x›
6377  by (rule RN) (metis (mono_tags, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(1) "modus-tollens:1" "o-objects-exist:1" "oa-contingent:2" "qml:2"[axiom_inst] "→E")
6378
6379AOT_theorem "o-objects-exist:5": ‹□¬∀x E!x›
6380proof (rule RN; rule "raa-cor:2")
6381  AOT_modally_strict {
6382    AOT_assume ‹∀x E!x›
6383    moreover AOT_obtain a where abs: ‹A!a›
6384      using "o-objects-exist:2"[THEN "qml:2"[axiom_inst, THEN "→E"]] "∃E"[rotated] by blast
6385    ultimately AOT_have ‹E!a› using "∀E" by blast
6386    AOT_hence 1: ‹◇E!a› by (metis "T◇" "→E")
6387    AOT_have ‹[λy ◇E!y]a›
6388    proof (rule "β←C"(1); "cqt:2[lambda]"?)
6389      AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst].
6390    next
6391      AOT_show ‹◇E!a› by (fact 1)
6392    qed
6393    AOT_hence ‹O!a›
6394      by (rule "=dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6395    AOT_hence ‹¬A!a› by (metis "≡E"(1) "oa-contingent:2") 
6396    AOT_thus ‹p & ¬p› for p using abs by (metis "raa-cor:3")
6397  }
6398qed
6399
6400AOT_theorem partition: ‹¬∃x (O!x & A!x)›
6401proof(rule "raa-cor:2")
6402  AOT_assume ‹∃x (O!x & A!x)›
6403  then AOT_obtain a where ‹O!a & A!a› using "∃E"[rotated] by blast
6404  AOT_thus ‹p & ¬p› for p by (metis "&E"(1) "Conjunction Simplification"(2) "≡E"(1) "modus-tollens:1" "oa-contingent:2" "raa-cor:3")
6405qed
6406
6407AOT_define eq_E :: ‹Π› ("'(=E')") "=E": ‹(=E) =df [λxy O!x & O!y & □∀F ([F]x ≡ [F]y)]›
6408
6409syntax "_AOT_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "=E" 50)
6410translations
6411  "_AOT_eq_E_infix κ κ'" == "CONST AOT_exe (CONST eq_E) (CONST Pair κ κ')"
6412(* TODO: try to replace by a simple translations pattern *)
6413print_translation‹
6414AOT_syntax_print_translations
6415[(const_syntax‹AOT_exe›, fn ctxt => fn [
6416  Const ("constAOT_PLM.eq_E", _),
6417  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6418] => Const (syntax_const‹_AOT_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6419
6420text‹Note: Not explicitly mentioned as theorem in PLM.›
6421AOT_theorem "=E[denotes]": ‹[(=E)]↓›
6422  by (rule "=dfI"(2)[OF "=E"]) "cqt:2[lambda]"+
6423
6424AOT_theorem "=E-simple:1": ‹x =E y ≡ (O!x & O!y & □∀F ([F]x ≡ [F]y))›
6425proof -
6426  (* TODO: rethink the product hacks *)
6427  AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6428    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6429  AOT_have 1: ‹[λxy [O!]x & [O!]y & □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6430  show ?thesis apply (rule "=dfI"(2)[OF "=E"]; "cqt:2[lambda]"?)
6431    using "beta-C-meta"[THEN "→E", OF 1, unvarify ν1νn, of "(AOT_term_of_var x,AOT_term_of_var y)", OF 0]
6432    by fast
6433qed
6434
6435AOT_theorem "=E-simple:2": ‹x =E y → x = y›
6436proof (rule "→I")
6437  AOT_assume ‹x =E y›
6438  AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)› using "=E-simple:1"[THEN "≡E"(1)] by blast
6439  AOT_thus ‹x = y›
6440    using "≡dfI"[OF "identity:1"] "∨I" by blast
6441qed
6442
6443AOT_theorem "id-nec3:1": ‹x =E y ≡ □(x =E y)›
6444proof (rule "≡I"; rule "→I")
6445  AOT_assume ‹x =E y›
6446  AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6447    using "=E-simple:1" "≡E" by blast
6448  AOT_hence ‹□O!x & □O!y & □□∀F ([F]x ≡ [F]y)›
6449    by (metis "S5Basic:6" "&I" "&E"(1) "&E"(2) "≡E"(4) "oa-facts:1" "raa-cor:3" "vdash-properties:10")
6450  AOT_hence ‹□(O!x & O!y & □∀F ([F]x ≡ [F]y))›
6451    by (metis "&E"(1) "&E"(2) "≡E"(2) "KBasic:3" "&I")
6452  AOT_thus ‹□(x =E y)›
6453    using "=E-simple:1"
6454    by (AOT_subst ‹x =E y› ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›) auto
6455next
6456  AOT_assume ‹□(x =E y)›
6457  AOT_thus ‹x =E y› using "qml:2"[axiom_inst, THEN "→E"] by blast
6458qed
6459
6460AOT_theorem "id-nec3:2": ‹◇(x =E y) ≡ x =E y›
6461  by (meson "RE◇" "S5Basic:2" "id-nec3:1" "≡E"(1) "≡E"(5) "Commutativity of ≡")
6462
6463AOT_theorem "id-nec3:3": ‹◇(x =E y) ≡ □(x =E y)›
6464  by (meson "id-nec3:1" "id-nec3:2" "≡E"(5))
6465
6466syntax "_AOT_non_eq_E" :: ‹Π› ("'(≠E')")
6467translations
6468  (Π) "(≠E)" == (Π) "(=E)-"
6469syntax "_AOT_non_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "≠E" 50)
6470translations
6471 "_AOT_non_eq_E_infix κ κ'" == "CONST AOT_exe (CONST relation_negation (CONST eq_E)) (CONST Pair κ κ')"
6472(* TODO: try replacing be a simple translations pattern *)
6473print_translation‹
6474AOT_syntax_print_translations
6475[(const_syntax‹AOT_exe›, fn ctxt => fn [
6476  Const (const_syntax‹relation_negation›, _) $ Const ("constAOT_PLM.eq_E", _),
6477  Const (const_syntax‹Pair›, _) $ lhs $ rhs
6478] => Const (syntax_const‹_AOT_non_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6479AOT_theorem "thm-neg=E": ‹x ≠E y ≡ ¬(x =E y)›
6480proof -
6481  (* TODO: rethink the product hacks *)
6482  AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6483    by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6484  AOT_have θ: ‹[λx1...x2 ¬(=E)x1...x2]↓› by "cqt:2[lambda]" (* TODO_PLM: convoluted proof in PLM; TODO: product hack *)
6485  AOT_have ‹x ≠E y ≡ [λx1...x2 ¬(=E)x1...x2]xy›
6486    by (rule "=dfI"(1)[OF "df-relation-negation", OF θ])
6487       (meson "oth-class-taut:3:a")
6488  also AOT_have ‹… ≡ ¬(=E)xy›
6489    apply (rule "beta-C-meta"[THEN "→E", unvarify ν1νn])
6490     apply "cqt:2[lambda]"
6491    by (fact 0)
6492  finally show ?thesis.
6493qed
6494
6495AOT_theorem "id-nec4:1": ‹x ≠E y ≡ □(x ≠E y)›
6496proof -
6497  AOT_have ‹x ≠E y ≡ ¬(x =E y)› using "thm-neg=E".
6498  also AOT_have ‹… ≡ ¬◇(x =E y)›
6499    by (meson "id-nec3:2" "≡E"(1) "Commutativity of ≡" "oth-class-taut:4:b")
6500  also AOT_have ‹… ≡ □¬(x =E y)›
6501    by (meson "KBasic2:1" "≡E"(2) "Commutativity of ≡")
6502  also AOT_have ‹… ≡ □(x ≠E y)›
6503    by (AOT_subst (reverse) ‹¬(x =E y)› ‹x ≠E y›)
6504       (auto simp: "thm-neg=E" "oth-class-taut:3:a")
6505  finally show ?thesis.
6506qed
6507
6508AOT_theorem "id-nec4:2": ‹◇(x ≠E y) ≡ (x ≠E y)›
6509  by (meson "RE◇" "S5Basic:2" "id-nec4:1" "≡E"(2) "≡E"(5) "Commutativity of ≡")
6510
6511AOT_theorem "id-nec4:3": ‹◇(x ≠E y) ≡ □(x ≠E y)›
6512  by (meson "id-nec4:1" "id-nec4:2" "≡E"(5))
6513
6514AOT_theorem "id-act2:1": ‹x =E y ≡ 𝒜x =E y›
6515  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec3:2" "≡E"(1) "≡E"(6))
6516AOT_theorem "id-act2:2": ‹x ≠E y ≡ 𝒜x ≠E y›
6517  by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec4:2" "≡E"(1) "≡E"(6))
6518
6519AOT_theorem "ord=Eequiv:1": ‹O!x → x =E x›
6520proof (rule "→I")
6521  AOT_assume 1: ‹O!x›
6522  AOT_show ‹x =E x›
6523    apply (rule "=dfI"(2)[OF "=E"]) apply "cqt:2[lambda]"
6524    apply (rule "β←C"(1))
6525      apply "cqt:2[lambda]"
6526     apply (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6527    by (simp add: "1" RN "&I" "oth-class-taut:3:a" "universal-cor")
6528qed
6529
6530AOT_theorem "ord=Eequiv:2": ‹x =E y → y =E x›
6531proof(rule CP)
6532  AOT_assume 1: ‹x =E y›
6533  AOT_hence 2: ‹x = y› by (metis "=E-simple:2" "vdash-properties:10") 
6534  AOT_have ‹O!x› using 1 by (meson "&E"(1) "=E-simple:1" "≡E"(1))
6535  AOT_hence ‹x =E x› using "ord=Eequiv:1" "→E" by blast
6536  AOT_thus ‹y =E x› using "rule=E"[rotated, OF 2] by fast
6537qed
6538
6539AOT_theorem "ord=Eequiv:3": ‹(x =E y & y =E z) → x =E z›
6540proof (rule CP)
6541  AOT_assume 1: ‹x =E y & y =E z›
6542  AOT_hence ‹x = y & y = z›
6543    by (metis "&I" "&E"(1) "&E"(2) "=E-simple:2" "vdash-properties:6")
6544  AOT_hence ‹x = z› by (metis "id-eq:3" "vdash-properties:6")
6545  moreover AOT_have ‹x =E x›
6546    using 1[THEN "&E"(1)] "&E"(1) "=E-simple:1" "≡E"(1) "ord=Eequiv:1" "→E" by blast
6547  ultimately AOT_show ‹x =E z›
6548    using "rule=E" by fast
6549qed
6550
6551AOT_theorem "ord-=E=:1": ‹(O!x ∨ O!y) → □(x = y ≡ x =E y)›
6552proof(rule CP)
6553  AOT_assume ‹O!x ∨ O!y›
6554  moreover {
6555    AOT_assume ‹O!x›
6556    AOT_hence ‹□O!x› by (metis "oa-facts:1" "vdash-properties:10")
6557    moreover {
6558      AOT_modally_strict {
6559        AOT_have ‹O!x → (x = y ≡ x =E y)›
6560        proof (rule "→I"; rule "≡I"; rule "→I")
6561          AOT_assume ‹O!x›
6562          AOT_hence ‹x =E x› by (metis "ord=Eequiv:1" "→E")
6563          moreover AOT_assume ‹x = y›
6564          ultimately AOT_show ‹x =E y› using "rule=E" by fast
6565        next
6566          AOT_assume ‹x =E y›
6567          AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6568        qed
6569      }
6570      AOT_hence ‹□O!x → □(x = y ≡ x =E y)› by (metis "RM:1")
6571    }
6572    ultimately AOT_have ‹□(x = y ≡ x =E y)› using "→E" by blast
6573  }
6574  moreover {
6575    AOT_assume ‹O!y›
6576    AOT_hence ‹□O!y› by (metis "oa-facts:1" "vdash-properties:10")
6577    moreover {
6578      AOT_modally_strict {
6579        AOT_have ‹O!y → (x = y ≡ x =E y)›
6580        proof (rule "→I"; rule "≡I"; rule "→I")
6581          AOT_assume ‹O!y›
6582          AOT_hence ‹y =E y› by (metis "ord=Eequiv:1" "→E")
6583          moreover AOT_assume ‹x = y›
6584          ultimately AOT_show ‹x =E y› using "rule=E" id_sym by fast
6585        next
6586          AOT_assume ‹x =E y›
6587          AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6588        qed
6589      }
6590      AOT_hence ‹□O!y → □(x = y ≡ x =E y)› by (metis "RM:1")
6591    }
6592    ultimately AOT_have ‹□(x = y ≡ x =E y)› using "→E" by blast
6593  }
6594  ultimately AOT_show ‹□(x = y ≡ x =E y)› by (metis "∨E"(3) "raa-cor:1")
6595qed
6596
6597AOT_theorem "ord-=E=:2": ‹O!y → [λx x = y]↓›
6598proof (rule "→I"; rule "safe-ext"[axiom_inst, THEN "→E"]; rule "&I")
6599  AOT_show ‹[λx x =E y]↓› by "cqt:2[lambda]"
6600next
6601  AOT_assume ‹O!y›
6602  AOT_hence 1: ‹□(x = y ≡ x =E y)› for x using "ord-=E=:1" "→E" "∨I" by blast
6603  AOT_have ‹□(x =E y ≡ x = y)› for x
6604    by (AOT_subst ‹x =E y ≡ x = y› ‹x = y ≡ x =E y›)
6605       (auto simp add: "Commutativity of ≡" 1)
6606  AOT_hence ‹∀x □(x =E y ≡ x = y)› by (rule GEN)
6607  AOT_thus ‹□∀x (x =E y ≡ x = y)› by (rule BF[THEN "→E"])
6608qed
6609
6610
6611AOT_theorem "ord-=E=:3": ‹[λxy O!x & O!y & x = y]↓›
6612proof (rule "safe-ext[2]"[axiom_inst, THEN "→E"]; rule "&I")
6613  AOT_show ‹[λxy O!x & O!y & x =E y]↓› by "cqt:2[lambda]"
6614next
6615  AOT_show ‹□∀x∀y ([O!]x & [O!]y & x =E y ≡ [O!]x & [O!]y & x = y)›
6616  proof (rule RN; rule GEN; rule GEN; rule "≡I"; rule "→I")
6617    AOT_modally_strict {
6618      AOT_show ‹[O!]x & [O!]y & x = y› if ‹[O!]x & [O!]y & x =E y› for x y
6619        by (metis "&I" "&E"(1) "Conjunction Simplification"(2) "=E-simple:2"
6620                  "modus-tollens:1" "raa-cor:1" that)
6621    }
6622  next
6623    AOT_modally_strict {
6624      AOT_show ‹[O!]x & [O!]y & x =E y› if ‹[O!]x & [O!]y & x = y› for x y
6625        apply(safe intro!: "&I")
6626          apply (metis that[THEN "&E"(1), THEN "&E"(1)])
6627         apply (metis that[THEN "&E"(1), THEN "&E"(2)])
6628        using "rule=E"[rotated, OF that[THEN "&E"(2)]]
6629              "ord=Eequiv:1"[THEN "→E", OF that[THEN "&E"(1), THEN "&E"(1)]] by fast
6630    }
6631  qed
6632qed
6633
6634AOT_theorem "ind-nec": ‹∀F ([F]x ≡ [F]y) → □∀F ([F]x ≡ [F]y)›
6635proof(rule "→I")
6636  AOT_assume ‹∀F ([F]x ≡ [F]y)›
6637  moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6638  ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x ≡ [λx □∀F ([F]x ≡ [F]y)]y›
6639    using "∀E" by blast
6640  moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]y›
6641    apply (rule "β←C"(1))
6642      apply "cqt:2[lambda]"
6643     apply (fact "cqt:2[const_var]"[axiom_inst])
6644    by (simp add: RN GEN "oth-class-taut:3:a")
6645  ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x› using "≡E" by blast
6646  AOT_thus ‹□∀F ([F]x ≡ [F]y)›
6647    using "β→C"(1) by blast
6648qed
6649
6650AOT_theorem "ord=E:1": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x =E y)›
6651proof (rule "→I"; rule "→I")
6652  AOT_assume ‹∀F ([F]x ≡ [F]y)›
6653  AOT_hence ‹□∀F ([F]x ≡ [F]y)›
6654    using "ind-nec"[THEN "→E"] by blast
6655  moreover AOT_assume ‹O!x & O!y›
6656  ultimately AOT_have ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6657    using "&I" by blast
6658  AOT_thus ‹x =E y› using "=E-simple:1"[THEN "≡E"(2)] by blast
6659qed
6660
6661AOT_theorem "ord=E:2": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x = y)›
6662proof (rule "→I"; rule "→I")
6663  AOT_assume ‹O!x & O!y›
6664  moreover AOT_assume ‹∀F ([F]x ≡ [F]y)›
6665  ultimately AOT_have ‹x =E y›
6666    using "ord=E:1" "→E" by blast
6667  AOT_thus ‹x = y› using "=E-simple:2"[THEN "→E"] by blast
6668qed
6669
6670AOT_theorem "ord=E2:1": ‹(O!x & O!y) → (x ≠ y ≡ [λz z =E x] ≠ [λz z =E y])›
6671proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6672  AOT_assume 0: ‹O!x & O!y›
6673  AOT_assume ‹x ≠ y›
6674  AOT_hence 1: ‹¬(x = y)› using "≡dfE"[OF "=-infix"] by blast
6675  AOT_assume ‹[λz z =E x] = [λz z =E y]›
6676  moreover AOT_have ‹[λz z =E x]x›
6677    apply (rule "β←C"(1))
6678      apply "cqt:2[lambda]"
6679     apply (fact "cqt:2[const_var]"[axiom_inst])
6680    using "ord=Eequiv:1"[THEN "→E", OF 0[THEN "&E"(1)]].
6681  ultimately AOT_have ‹[λz z =E y]x› using "rule=E" by fast
6682  AOT_hence ‹x =E y› using "β→C"(1) by blast
6683  AOT_hence ‹x = y› by (metis "=E-simple:2" "vdash-properties:6")
6684  AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6685next
6686  AOT_assume ‹[λz z =E x] ≠ [λz z =E y]›
6687  AOT_hence 0: ‹¬([λz z =E x] = [λz z =E y])› using "≡dfE"[OF "=-infix"] by blast
6688  AOT_have ‹[λz z =E x]↓› by "cqt:2[lambda]"
6689  AOT_hence ‹[λz z =E x] = [λz z =E x]›
6690    by (metis "rule=I:1")
6691  moreover AOT_assume ‹x = y›
6692  ultimately AOT_have ‹[λz z =E x] = [λz z =E y]›
6693    using "rule=E" by fast
6694  AOT_thus ‹[λz z =E x] = [λz z =E y] & ¬([λz z =E x] = [λz z =E y])›
6695    using 0 "&I" by blast
6696qed
6697
6698AOT_theorem "ord=E2:2": ‹(O!x & O!y) → (x ≠ y ≡ [λz z = x] ≠ [λz z = y])›
6699proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6700  AOT_assume 0: ‹O!x & O!y›
6701  AOT_assume ‹x ≠ y›
6702  AOT_hence 1: ‹¬(x = y)› using "≡dfE"[OF "=-infix"] by blast
6703  AOT_assume ‹[λz z = x] = [λz z = y]›
6704  moreover AOT_have ‹[λz z = x]x›
6705    apply (rule "β←C"(1))
6706    apply (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6707     apply (fact "cqt:2[const_var]"[axiom_inst])
6708    by (simp add: "id-eq:1")
6709  ultimately AOT_have ‹[λz z = y]x› using "rule=E" by fast
6710  AOT_hence ‹x = y› using "β→C"(1) by blast
6711  AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6712next
6713  AOT_assume 0: ‹O!x & O!y›
6714  AOT_assume ‹[λz z = x] ≠ [λz z = y]›
6715  AOT_hence 1: ‹¬([λz z = x] = [λz z = y])› using "≡dfE"[OF "=-infix"] by blast
6716  AOT_have ‹[λz z = x]↓› by (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6717  AOT_hence ‹[λz z = x] = [λz z = x]›
6718    by (metis "rule=I:1")
6719  moreover AOT_assume ‹x = y›
6720  ultimately AOT_have ‹[λz z = x] = [λz z = y]›
6721    using "rule=E" by fast
6722  AOT_thus ‹[λz z = x] = [λz z = y] & ¬([λz z = x] = [λz z = y])›
6723    using 1 "&I" by blast
6724qed
6725
6726AOT_theorem ordnecfail: ‹O!x → □¬∃F x[F]›
6727  by (meson "RM:1" "deduction-theorem" nocoder "oa-facts:1" "vdash-properties:10" "vdash-properties:1[2]")
6728
6729AOT_theorem "ab-obey:1": ‹(A!x & A!y) → (∀F (x[F] ≡ y[F]) → x = y)›
6730proof (rule "→I"; rule "→I")
6731  AOT_assume 1: ‹A!x & A!y›
6732  AOT_assume ‹∀F (x[F] ≡ y[F])›
6733  AOT_hence ‹x[F] ≡ y[F]› for F using "∀E" by blast
6734  AOT_hence ‹□(x[F] ≡ y[F])› for F by (metis "en-eq:6[1]" "≡E"(1))
6735  AOT_hence ‹∀F □(x[F] ≡ y[F])› by (rule GEN)
6736  AOT_hence ‹□∀F (x[F] ≡ y[F])› by (rule BF[THEN "→E"])
6737  AOT_thus ‹x = y›
6738    using "≡dfI"[OF "identity:1", OF "∨I"(2)] 1 "&I" by blast
6739qed
6740
6741AOT_theorem "ab-obey:2": ‹(∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])) → x ≠ y›
6742proof (rule "→I"; rule "≡dfI"[OF "=-infix"]; rule "raa-cor:2")
6743  AOT_assume 1: ‹x = y›
6744  AOT_assume ‹∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])›
6745  moreover {
6746    AOT_assume ‹∃F (x[F] & ¬y[F])›
6747    then AOT_obtain F where ‹x[F] & ¬y[F]› using "∃E"[rotated] by blast
6748    moreover AOT_have ‹y[F]› using calculation[THEN "&E"(1)] 1 "rule=E" by fast
6749    ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(2) "modus-tollens:2" "raa-cor:3")
6750  }
6751  moreover {
6752    AOT_assume ‹∃F (y[F] & ¬x[F])›
6753    then AOT_obtain F where ‹y[F] & ¬x[F]› using "∃E"[rotated] by blast
6754    moreover AOT_have ‹¬y[F]› using calculation[THEN "&E"(2)] 1 "rule=E" by fast
6755    ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(1) "modus-tollens:1" "raa-cor:3")
6756  }
6757  ultimately AOT_show ‹p & ¬p› for p by (metis "∨E"(3) "raa-cor:1")
6758qed
6759
6760AOT_theorem "encoders-are-abstract": ‹∃F x[F] → A!x›
6761  by (meson "deduction-theorem" "≡E"(2) "modus-tollens:2" nocoder
6762            "oa-contingent:3" "vdash-properties:1[2]")
6763
6764AOT_theorem "denote=:1": ‹∀H∃x x[H]›
6765  by (rule GEN; rule "existence:2[1]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6766
6767AOT_theorem "denote=:2": ‹∀G∃x1...∃xn x1...xn[H]›
6768  by (rule GEN; rule "existence:2"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6769
6770AOT_theorem "denote=:2[2]": ‹∀G∃x1∃x2 x1x2[H]›
6771  by (rule GEN; rule "existence:2[2]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6772
6773AOT_theorem "denote=:2[3]": ‹∀G∃x1∃x2∃x3 x1x2x3[H]›
6774  by (rule GEN; rule "existence:2[3]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6775
6776AOT_theorem "denote=:2[4]": ‹∀G∃x1∃x2∃x3∃x4 x1x2x3x4[H]›
6777  by (rule GEN; rule "existence:2[4]"[THEN "≡dfE"]; fact "cqt:2[const_var]"[axiom_inst])
6778
6779AOT_theorem "denote=:3": ‹∃x x[Π] ≡ ∃H (H = Π)›
6780  using "existence:2[1]" "free-thms:1" "≡E"(2) "≡E"(5) "Commutativity of ≡" "≡Df" by blast
6781
6782AOT_theorem "denote=:4": ‹(∃x1...∃xn x1...xn[Π]) ≡ ∃H (H = Π)›
6783  using "existence:2" "free-thms:1" "≡E"(6) "≡Df" by blast
6784
6785AOT_theorem "denote=:4[2]": ‹(∃x1∃x2 x1x2[Π]) ≡ ∃H (H = Π)›
6786  using "existence:2[2]" "free-thms:1" "≡E"(6) "≡Df" by blast
6787
6788AOT_theorem "denote=:4[3]": ‹(∃x1∃x2∃x3 x1x2x3[Π]) ≡ ∃H (H = Π)›
6789  using "existence:2[3]" "free-thms:1" "≡E"(6) "≡Df" by blast
6790
6791AOT_theorem "denote=:4[4]": ‹(∃x1∃x2∃x3∃x4 x1x2x3x4[Π]) ≡ ∃H (H = Π)›
6792  using "existence:2[4]" "free-thms:1" "≡E"(6) "≡Df" by blast
6793
6794AOT_theorem "A-objects!": ‹∃!x (A!x & ∀F (x[F] ≡ φ{F}))›
6795proof (rule "uniqueness:1"[THEN "≡dfI"])
6796  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ φ{F})›
6797    using "A-objects"[axiom_inst] "∃E"[rotated] by blast
6798  AOT_have ‹(A!β & ∀F (β[F] ≡ φ{F})) → β = a› for β
6799  proof (rule "→I")
6800    AOT_assume β_prop: ‹[A!]β & ∀F (β[F] ≡ φ{F})›
6801    AOT_hence ‹β[F] ≡ φ{F}› for F using "∀E" "&E" by blast
6802    AOT_hence ‹β[F] ≡ a[F]› for F
6803      using a_prop[THEN "&E"(2)] "∀E" "≡E"(2) "≡E"(5) "Commutativity of ≡" by fast
6804    AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6805    AOT_thus ‹β = a›
6806      using "ab-obey:1"[THEN "→E", OF "&I"[OF β_prop[THEN "&E"(1)], OF a_prop[THEN "&E"(1)]], THEN "→E"] by blast
6807  qed
6808  AOT_hence ‹∀β ((A!β & ∀F (β[F] ≡ φ{F})) → β = a)› by (rule GEN)
6809  AOT_thus ‹∃α ([A!]α & ∀F (α[F] ≡ φ{F}) & ∀β ([A!]β & ∀F (β[F] ≡ φ{F}) → β = α)) ›
6810    using "∃I" using a_prop "&I" by fast
6811qed
6812
6813AOT_theorem "obj-oth:1": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y))›
6814  using "A-objects!" by fast
6815
6816AOT_theorem "obj-oth:2": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y & [F]z))›
6817  using "A-objects!" by fast
6818
6819AOT_theorem "obj-oth:3": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y ∨ [F]z))›
6820  using "A-objects!" by fast
6821
6822AOT_theorem "obj-oth:4": ‹∃!x (A!x & ∀F (x[F] ≡ □[F]y))›
6823  using "A-objects!" by fast
6824
6825AOT_theorem "obj-oth:5": ‹∃!x (A!x & ∀F (x[F] ≡ F = G))›
6826  using "A-objects!" by fast
6827
6828AOT_theorem "obj-oth:6": ‹∃!x (A!x & ∀F (x[F] ≡ □∀y([G]y → [F]y)))›
6829  using "A-objects!" by fast
6830
6831AOT_theorem "A-descriptions": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))↓›
6832  by (rule "A-Exists:2"[THEN "≡E"(2)]; rule "RA[2]"; rule "A-objects!")
6833
6834AOT_act_theorem "thm-can-terms2": ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6835  using "y-in:2" by blast
6836
6837AOT_theorem "can-ab2": ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) →  A!y›
6838proof(rule "→I")
6839  AOT_assume ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6840  AOT_hence ‹𝒜(A!y & ∀F (y[F] ≡ φ{F}))›
6841    using "actual-desc:2"[THEN "→E"] by blast
6842  AOT_hence ‹𝒜A!y› by (metis "Act-Basic:2" "&E"(1) "≡E"(1))
6843  AOT_thus ‹A!y› by (metis "≡E"(2) "oa-facts:8")
6844qed
6845
6846AOT_act_theorem "desc-encode:1": ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6847proof -
6848  AOT_have ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6849    by (simp add: "A-descriptions")
6850  AOT_hence ‹A!ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6851    using "y-in:3"[THEN "→E"] by blast
6852  AOT_thus ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6853    using "&E" "∀E" by blast
6854qed
6855
6856AOT_act_theorem "desc-encode:2": ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6857  using "desc-encode:1".
6858
6859AOT_theorem "desc-nec-encode:1": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6860proof -
6861  AOT_have 0: ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6862    by (simp add: "A-descriptions")
6863  AOT_hence ‹𝒜(A!ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}))›
6864    using "actual-desc:4"[THEN "→E"] by blast
6865  AOT_hence ‹𝒜∀F (ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6866    using "Act-Basic:2" "&E"(2) "≡E"(1) by blast
6867  AOT_hence ‹∀F 𝒜(ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6868    using "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]" by blast
6869  AOT_hence ‹𝒜(ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6870    using "∀E" by blast
6871  AOT_hence ‹𝒜ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6872    using "Act-Basic:5" "≡E"(1) by blast
6873  AOT_thus ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ 𝒜φ{F}›
6874    using "en-eq:10[1]"[unvarify x1, OF 0] "≡E"(6) by blast
6875qed
6876
6877AOT_theorem "desc-nec-encode:2": ‹ιx (A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ 𝒜φ{G}›
6878  using "desc-nec-encode:1".
6879
6880AOT_theorem "Box-desc-encode:1": ‹□φ{G} → ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]›
6881  by (rule "→I"; rule "desc-nec-encode:2"[THEN "≡E"(2)])
6882     (meson "nec-imp-act" "vdash-properties:10")
6883
6884AOT_theorem "Box-desc-encode:2": ‹□φ{G} → □(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6885proof(rule CP)
6886  AOT_assume ‹□φ{G}›
6887  AOT_hence ‹□□φ{G}› by (metis "S5Basic:6" "≡E"(1))
6888  moreover AOT_have ‹□□φ{G} → □(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6889  proof (rule RM; rule "→I")
6890    AOT_modally_strict {
6891      AOT_assume 1: ‹□φ{G}›
6892      AOT_hence ‹ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]› using "Box-desc-encode:1" "→E" by blast
6893      moreover AOT_have ‹φ{G}› using 1 by (meson "qml:2" "vdash-properties:10" "vdash-properties:1[2]")
6894      ultimately AOT_show ‹ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G}›
6895        using "deduction-theorem" "≡I" by simp
6896    }
6897  qed
6898  ultimately AOT_show ‹□(ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})› using "→E" by blast
6899qed
6900
6901definition rigid_condition where ‹rigid_condition φ ≡ ∀v . [v ⊨ ∀α (φ{α} → □φ{α})]›
6902syntax rigid_condition :: ‹id_position ⇒ AOT_prop› ("RIGID'_CONDITION'(_')")
6903
6904AOT_theorem "strict-can:1[E]": assumes ‹RIGID_CONDITION(φ)›
6905  shows ‹∀α (φ{α} → □φ{α})›
6906  using assms[unfolded rigid_condition_def] by auto
6907
6908AOT_theorem "strict-can:1[I]":
6909  assumes ‹⊢□ ∀α (φ{α} → □φ{α})›
6910  shows ‹RIGID_CONDITION(φ)›
6911  using assms rigid_condition_def by auto
6912
6913AOT_theorem "box-phi-a:1": assumes ‹RIGID_CONDITION(φ)›
6914  shows ‹(A!x  & ∀F (x[F] ≡ φ{F})) → □(A!x & ∀F (x[F] ≡ φ{F}))›
6915proof (rule "→I")
6916  AOT_assume a: ‹A!x & ∀F (x[F] ≡ φ{F})›
6917  AOT_hence b: ‹□A!x› by (metis "Conjunction Simplification"(1) "oa-facts:2" "vdash-properties:10")
6918  AOT_have ‹x[F] ≡ φ{F}› for F using a[THEN "&E"(2)] "∀E" by blast
6919  moreover AOT_have ‹□(x[F] → □x[F])› for F by (meson "pre-en-eq:1[1]" RN)
6920  moreover AOT_have ‹□(φ{F} → □φ{F})› for F using RN "strict-can:1[E]"[OF assms] "∀E" by blast
6921  ultimately AOT_have ‹□(x[F] ≡ φ{F})› for F
6922    using "sc-eq-box-box:5" "qml:2"[axiom_inst, THEN "→E"] "→E" "&I" by metis
6923  AOT_hence ‹∀F □(x[F] ≡ φ{F})› by (rule GEN)
6924  AOT_hence ‹□∀F (x[F] ≡ φ{F})› by (rule BF[THEN "→E"])
6925  AOT_thus ‹□([A!]x & ∀F (x[F] ≡ φ{F}))›
6926    using b "KBasic:3" "≡S"(1) "≡E"(2) by blast
6927qed
6928
6929AOT_theorem "box-phi-a:2": assumes ‹RIGID_CONDITION(φ)›
6930  shows ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6931proof(rule "→I")
6932  AOT_assume ‹y = ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6933  AOT_hence ‹𝒜(A!y & ∀F (y[F] ≡ φ{F}))› using "actual-desc:2"[THEN "→E"] by fast
6934  AOT_hence abs: ‹𝒜A!y› and ‹𝒜∀F (y[F] ≡ φ{F})›
6935    using "Act-Basic:2" "&E" "≡E"(1) by blast+
6936  AOT_hence ‹∀F 𝒜(y[F] ≡ φ{F})› by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
6937  AOT_hence ‹𝒜(y[F] ≡ φ{F})› for F using "∀E" by blast
6938  AOT_hence ‹𝒜y[F] ≡ 𝒜φ{F}› for F by (metis "Act-Basic:5" "≡E"(1)) 
6939  AOT_hence ‹y[F] ≡ φ{F}› for F
6940    using "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6941    by (metis "en-eq:10[1]" "≡E"(6))
6942  AOT_hence ‹∀F (y[F] ≡ φ{F})› by (rule GEN)
6943  AOT_thus ‹[A!]y & ∀F (y[F] ≡ φ{F})› using abs "&I" "≡E"(2) "oa-facts:8" by blast
6944qed
6945
6946AOT_theorem "box-phi-a:3": assumes ‹RIGID_CONDITION(φ)›
6947  shows ‹ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6948  using "desc-nec-encode:2"
6949    "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6950    "≡E"(5) by blast
6951
6952AOT_define Null :: ‹τ ⇒ φ› ("Null'(_')") 
6953  "df-null-uni:1": ‹Null(x) ≡df A!x & ¬∃F x[F]›
6954
6955AOT_define Universal :: ‹τ ⇒ φ› ("Universal'(_')")
6956  "df-null-uni:2": ‹Universal(x) ≡df A!x & ∀F x[F]›
6957
6958AOT_theorem "null-uni-uniq:1": ‹∃!x Null(x)›
6959proof (rule "uniqueness:1"[THEN "≡dfI"])
6960  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ¬(F = F))›
6961    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6962  AOT_have a_null: ‹¬a[F]› for F
6963  proof (rule "raa-cor:2")
6964    AOT_assume ‹a[F]›
6965    AOT_hence ‹¬(F = F)› using a_prop[THEN "&E"(2)] "∀E" "≡E" by blast
6966    AOT_hence ‹F = F & ¬(F = F)› by (metis "id-eq:1" "raa-cor:3")
6967    AOT_thus ‹p & ¬p› for p  by (metis "raa-cor:1")
6968  qed
6969  AOT_have ‹Null(a) & ∀β (Null(β) → β = a)›
6970  proof (rule "&I")
6971    AOT_have ‹¬∃F a[F]› using a_null by (metis "instantiation" "reductio-aa:1")
6972    AOT_thus ‹Null(a)›
6973      using "df-null-uni:1"[THEN "≡dfI"] a_prop[THEN "&E"(1)] "&I" by metis
6974  next
6975    AOT_show ‹∀β (Null(β) → β = a)›
6976    proof (rule GEN; rule "→I")
6977      fix β
6978      AOT_assume a: ‹Null(β)›
6979      AOT_hence ‹¬∃F β[F]›
6980        using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
6981      AOT_hence β_null: ‹¬β[F]› for F by (metis "existential:2[const_var]" "reductio-aa:1")
6982      AOT_have ‹∀F (β[F] ≡ a[F])›
6983        apply (rule GEN; rule "≡I"; rule CP)
6984        using "raa-cor:3" β_null a_null by blast+
6985      moreover AOT_have ‹A!β› using a "df-null-uni:1"[THEN "≡dfE"] "&E" by blast
6986      ultimately AOT_show ‹β = a›
6987        using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" by blast
6988    qed
6989  qed
6990  AOT_thus ‹∃α (Null(α) & ∀β (Null(β) → β = α))› using "∃I"(2) by fast
6991qed
6992
6993AOT_theorem "null-uni-uniq:2": ‹∃!x Universal(x)›
6994proof (rule "uniqueness:1"[THEN "≡dfI"])
6995  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ F = F)›
6996    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6997  AOT_hence aF: ‹a[F]› for F using "&E" "∀E" "≡E" "id-eq:1" by fast
6998  AOT_hence ‹Universal(a)›
6999    using "df-null-uni:2"[THEN "≡dfI"] "&I" a_prop[THEN "&E"(1)] GEN by blast
7000  moreover AOT_have ‹∀β (Universal(β) → β = a)›
7001  proof (rule GEN; rule "→I")
7002    fix β
7003    AOT_assume ‹Universal(β)›
7004    AOT_hence abs_β: ‹A!β› and ‹β[F]› for F using "df-null-uni:2"[THEN "≡dfE"] "&E" "∀E" by blast+
7005    AOT_hence ‹β[F] ≡ a[F]› for F using aF by (metis "deduction-theorem" "≡I")
7006    AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
7007    AOT_thus ‹β = a›
7008      using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" abs_β by blast
7009  qed
7010  ultimately AOT_show ‹∃α (Universal(α) & ∀β (Universal(β) → β = α))›
7011    using "&I" "∃I" by fast
7012qed
7013
7014AOT_theorem "null-uni-uniq:3": ‹ιx Null(x)↓›
7015  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:1" by blast
7016
7017AOT_theorem "null-uni-uniq:4": ‹ιx Universal(x)↓›
7018  using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:2" by blast
7019
7020AOT_define Null_object :: ‹κs› (‹a∅›)
7021  "df-null-uni-terms:1": ‹a∅ =df ιx Null(x)›
7022
7023AOT_define Universal_object :: ‹κs› (‹aV›)
7024  "df-null-uni-terms:2": ‹aV =df ιx Universal(x)›
7025
7026AOT_theorem "null-uni-facts:1": ‹Null(x) → □Null(x)›
7027proof (rule "→I")
7028  AOT_assume ‹Null(x)›
7029  AOT_hence x_abs: ‹A!x› and x_null: ‹¬∃F x[F]›
7030    using "df-null-uni:1"[THEN "≡dfE"] "&E" by blast+
7031  AOT_have ‹¬x[F]› for F using x_null
7032    using "existential:2[const_var]" "reductio-aa:1"
7033    by metis
7034  AOT_hence ‹□¬x[F]› for F by (metis "en-eq:7[1]" "≡E"(1))
7035  AOT_hence ‹∀F □¬x[F]› by (rule GEN)
7036  AOT_hence ‹□∀F ¬x[F]› by (rule BF[THEN "→E"])
7037  moreover AOT_have ‹□∀F ¬x[F] → □¬∃F x[F]›
7038    apply (rule RM)
7039    by (metis (full_types) "instantiation" "cqt:2[const_var]" "deduction-theorem"
7040                           "reductio-aa:1" "rule-ui:1" "vdash-properties:1[2]")
7041  ultimately AOT_have ‹□¬∃F x[F]›
7042    by (metis "→E")
7043  moreover AOT_have ‹□A!x› using x_abs
7044    using "oa-facts:2" "vdash-properties:10" by blast
7045  ultimately AOT_have r: ‹□(A!x & ¬∃F x[F])›
7046    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
7047  AOT_show ‹□Null(x)›
7048    by (AOT_subst ‹Null(x)› ‹A!x & ¬∃F x[F]›)
7049       (auto simp: "df-null-uni:1" "≡Df" r)
7050qed  
7051
7052AOT_theorem "null-uni-facts:2": ‹Universal(x) → □Universal(x)›
7053proof (rule "→I")
7054  AOT_assume ‹Universal(x)›
7055  AOT_hence x_abs: ‹A!x› and x_univ: ‹∀F x[F]›
7056    using "df-null-uni:2"[THEN "≡dfE"] "&E" by blast+
7057  AOT_have ‹x[F]› for F using x_univ "∀E" by blast
7058  AOT_hence ‹□x[F]› for F by (metis "en-eq:2[1]" "≡E"(1))
7059  AOT_hence ‹∀F □x[F]› by (rule GEN)
7060  AOT_hence ‹□∀F x[F]› by (rule BF[THEN "→E"])
7061  moreover AOT_have ‹□A!x› using x_abs
7062    using "oa-facts:2" "vdash-properties:10" by blast
7063  ultimately AOT_have r: ‹□(A!x & ∀F x[F])›
7064    by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
7065  AOT_show ‹□Universal(x)›
7066    by (AOT_subst ‹Universal(x)› ‹A!x & ∀F x[F]›)
7067       (auto simp add: "df-null-uni:2" "≡Df" r)
7068qed
7069
7070AOT_theorem "null-uni-facts:3": ‹Null(a∅)›
7071  apply (rule "=dfI"(2)[OF "df-null-uni-terms:1"])
7072   apply (simp add: "null-uni-uniq:3")
7073  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:3"]
7074    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:1"[unvarify x, THEN RN, OF "null-uni-uniq:3"], THEN "≡E"(1)]
7075  by blast
7076
7077AOT_theorem "null-uni-facts:4": ‹Universal(aV)›
7078  apply (rule "=dfI"(2)[OF "df-null-uni-terms:2"])
7079   apply (simp add: "null-uni-uniq:4")
7080  using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:4"]
7081    "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:2"[unvarify x, THEN RN, OF "null-uni-uniq:4"], THEN "≡E"(1)]
7082  by blast
7083
7084AOT_theorem "null-uni-facts:5": ‹a∅ ≠ aV›
7085proof (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
7086    rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
7087    rule "≡dfI"[OF "=-infix"];
7088    rule "raa-cor:2")
7089  AOT_obtain x where nullx: ‹Null(x)›
7090    by (metis "instantiation" "df-null-uni-terms:1" "existential:1" "null-uni-facts:3"
7091              "null-uni-uniq:3" "rule-id-df:2:b[zero]")
7092  AOT_hence act_null: ‹𝒜Null(x)› by (metis "nec-imp-act" "null-uni-facts:1" "vdash-properties:10")
7093  AOT_assume ‹ιx Null(x) = ιx Universal(x)›
7094  AOT_hence ‹𝒜∀x(Null(x) ≡ Universal(x))›
7095    using "actual-desc:5"[THEN "→E"] by blast
7096  AOT_hence ‹∀x 𝒜(Null(x) ≡ Universal(x))›
7097    by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
7098  AOT_hence ‹𝒜Null(x) ≡ 𝒜Universal(x)›
7099    using "Act-Basic:5" "≡E"(1) "rule-ui:3" by blast
7100  AOT_hence ‹𝒜Universal(x)› using act_null "≡E" by blast
7101  AOT_hence ‹Universal(x)› by (metis RN "≡E"(1) "null-uni-facts:2" "sc-eq-fur:2" "vdash-properties:10")
7102  AOT_hence ‹∀F x[F]› using "≡dfE"[OF "df-null-uni:2"] "&E" by metis
7103  moreover AOT_have ‹¬∃F x[F]› using nullx "≡dfE"[OF "df-null-uni:1"] "&E" by metis
7104  ultimately AOT_show ‹p & ¬p› for p by (metis "cqt-further:1" "raa-cor:3" "vdash-properties:10")
7105qed
7106
7107AOT_theorem "null-uni-facts:6": ‹a∅ = ιx(A!x & ∀F (x[F] ≡ F ≠ F))›
7108proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7109  AOT_show ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7110    by (simp add: "A-descriptions")
7111next
7112  AOT_show ‹a∅↓›
7113    by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7114       (simp add: "null-uni-uniq:3")
7115next
7116  AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7117    by (simp add: "A-descriptions")
7118  AOT_hence 1: ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F)) = ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7119    using "rule=I:1" by blast
7120  AOT_show ‹[A!]a∅ & [A!]ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7121    apply (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"]; rule "&I")
7122    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3" "null-uni-uniq:3" "rule-id-df:2:a[zero]" "vdash-properties:10")
7123    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7124next
7125  AOT_show ‹∀F (a∅[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F])›
7126  proof (rule GEN)
7127    fix F
7128    AOT_have ‹¬a∅[F]›
7129      by (rule "=dfI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7130         (metis (no_types, lifting) "≡dfE" "&E"(2) "∨I"(2) "∨E"(3)
7131                "df-null-uni:1" "df-null-uni-terms:1" "existential:2[const_var]" "null-uni-facts:3"
7132                "raa-cor:2" "rule-id-df:2:a[zero]" "russell-axiom[enc,1].ψ_denotes_asm")
7133    moreover AOT_have ‹¬ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7134    proof(rule "raa-cor:2")
7135      AOT_assume 0: ‹ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7136      AOT_hence ‹𝒜(F ≠ F)› using "desc-nec-encode:2"[THEN "≡E"(1), OF 0] by blast
7137      moreover AOT_have ‹¬𝒜(F ≠ F)›
7138        using "≡dfE" "id-act:2" "id-eq:1" "≡E"(2) "=-infix" "raa-cor:3" by blast
7139      ultimately AOT_show ‹𝒜(F ≠ F) & ¬𝒜(F ≠ F)› by (rule "&I")
7140    qed
7141    ultimately AOT_show ‹a∅[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7142      using "deduction-theorem" "≡I" "raa-cor:4" by blast
7143  qed
7144qed
7145
7146AOT_theorem "null-uni-facts:7": ‹aV = ιx(A!x & ∀F (x[F] ≡ F = F))›
7147proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7148  AOT_show ‹ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7149    by (simp add: "A-descriptions")
7150next
7151  AOT_show ‹aV↓›
7152    by (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7153       (simp add: "null-uni-uniq:4")
7154next
7155  AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7156    by (simp add: "A-descriptions")
7157  AOT_hence 1: ‹ιx([A!]x & ∀F (x[F] ≡ F = F)) = ιx([A!]x & ∀F (x[F] ≡ F = F))›
7158    using "rule=I:1" by blast
7159  AOT_show ‹[A!]aV & [A!]ιx([A!]x & ∀F (x[F] ≡ F = F))›
7160    apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"]; rule "&I")
7161    apply (meson "≡dfE" "Conjunction Simplification"(1) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "vdash-properties:10")
7162    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7163next
7164  AOT_show ‹∀F (aV[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F = F))[F])›
7165  proof (rule GEN)
7166    fix F
7167    AOT_have ‹aV[F]›
7168      apply (rule "=dfI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7169      using "≡dfE" "&E"(2) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "rule-ui:3" by blast
7170    moreover AOT_have ‹ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7171      using "RA[2]" "desc-nec-encode:2" "id-eq:1" "≡E"(2) by fastforce
7172    ultimately AOT_show ‹aV[F] ≡ ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7173      using "deduction-theorem" "≡I" by simp
7174  qed
7175qed
7176
7177AOT_theorem "aclassical:1": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])›
7178proof(rule GEN)
7179  fix R
7180  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]zy] & ¬y[F]))›
7181    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7182  AOT_have a_enc: ‹a[λz [R]za]›
7183  proof (rule "raa-cor:1")
7184    AOT_assume 0: ‹¬a[λz [R]za]›
7185    AOT_hence ‹¬∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7186      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]za]»"],
7187                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7188         "cqt:2[lambda]"
7189    AOT_hence ‹∀y ¬(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7190      using "cqt-further:4" "vdash-properties:10" by blast
7191    AOT_hence ‹¬(A!a & [λz [R]za] = [λz [R]za] & ¬a[λz [R]za])› using "∀E" by blast
7192    AOT_hence ‹(A!a & [λz [R]za] = [λz [R]za]) → a[λz [R]za]›
7193      by (metis "&I" "deduction-theorem" "raa-cor:3")
7194    moreover AOT_have ‹[λz [R]za] = [λz [R]za]›
7195      by (rule "=I") "cqt:2[lambda]"
7196    ultimately AOT_have ‹a[λz [R]za]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7197    AOT_thus ‹a[λz [R]za] & ¬a[λz [R]za]›
7198      using 0 "&I" by blast
7199  qed
7200  AOT_hence ‹∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7201    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7202  then AOT_obtain b where b_prop: ‹A!b & [λz [R]za] = [λz [R]zb] & ¬b[λz [R]za]›
7203    using "∃E"[rotated] by blast
7204  AOT_have ‹a ≠ b›
7205    apply (rule "≡dfI"[OF "=-infix"])
7206    using a_enc b_prop[THEN "&E"(2)]
7207    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7208  AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]za] = [λz [R]zb]›
7209    using b_prop "&E" a_prop "&I" by meson
7210  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]za] = [λz [R]zy])› by (rule "∃I")
7211  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])› by (rule "∃I")
7212qed
7213
7214AOT_theorem "aclassical:2": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])›
7215proof(rule GEN)
7216  fix R
7217  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]yz] & ¬y[F]))›
7218    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7219  AOT_have a_enc: ‹a[λz [R]az]›
7220  proof (rule "raa-cor:1")
7221    AOT_assume 0: ‹¬a[λz [R]az]›
7222    AOT_hence ‹¬∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7223      by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]az]»"],
7224                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7225         "cqt:2[lambda]"
7226    AOT_hence ‹∀y ¬(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7227      using "cqt-further:4" "vdash-properties:10" by blast
7228    AOT_hence ‹¬(A!a & [λz [R]az] = [λz [R]az] & ¬a[λz [R]az])› using "∀E" by blast
7229    AOT_hence ‹(A!a & [λz [R]az] = [λz [R]az]) → a[λz [R]az]›
7230      by (metis "&I" "deduction-theorem" "raa-cor:3")
7231    moreover AOT_have ‹[λz [R]az] = [λz [R]az]›
7232      by (rule "=I") "cqt:2[lambda]"
7233    ultimately AOT_have ‹a[λz [R]az]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7234    AOT_thus ‹a[λz [R]az] & ¬a[λz [R]az]›
7235      using 0 "&I" by blast
7236  qed
7237  AOT_hence ‹∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7238    by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7239  then AOT_obtain b where b_prop: ‹A!b & [λz [R]az] = [λz [R]bz] & ¬b[λz [R]az]›
7240    using "∃E"[rotated] by blast
7241  AOT_have ‹a ≠ b›
7242    apply (rule "≡dfI"[OF "=-infix"])
7243    using a_enc b_prop[THEN "&E"(2)]
7244    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7245  AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]az] = [λz [R]bz]›
7246    using b_prop "&E" a_prop "&I" by meson
7247  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]az] = [λz [R]yz])› by (rule "∃I")
7248  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])› by (rule "∃I")
7249qed
7250
7251AOT_theorem "aclassical:3": ‹∀F∃x∃y(A!x & A!y & x ≠ y & [λ [F]x] = [λ [F]y])›
7252proof(rule GEN)
7253  fix R
7254  AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]y] & ¬y[F]))›
7255    using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7256  AOT_have ‹[λz [R]a]↓› by "cqt:2[lambda]"
7257  (* TODO: S should no longer be necessary *)
7258  then AOT_obtain S where S_def: ‹S = [λz [R]a]›
7259    by (metis "instantiation" "rule=I:1" "existential:1" id_sym)
7260  AOT_have a_enc: ‹a[S]›
7261  proof (rule "raa-cor:1")
7262    AOT_assume 0: ‹¬a[S]›
7263    AOT_hence ‹¬∃y(A!y & S = [λz [R]y] & ¬y[S])›
7264      by (rule a_prop[THEN "&E"(2), THEN "∀E"(2)[where β=S],
7265                THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated]) 
7266    AOT_hence ‹∀y ¬(A!y & S = [λz [R]y] & ¬y[S])›
7267      using "cqt-further:4" "vdash-properties:10" by blast
7268    AOT_hence ‹¬(A!a & S = [λz [R]a] & ¬a[S])› using "∀E" by blast
7269    AOT_hence ‹(A!a & S = [λz [R]a]) → a[S]›
7270      by (metis "&I" "deduction-theorem" "raa-cor:3")
7271    moreover AOT_have ‹S = [λz [R]a]› using S_def .
7272    ultimately AOT_have ‹a[S]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7273    AOT_thus ‹a[λz [R]a] & ¬a[λz [R]a]›  by (metis "0" "raa-cor:3") 
7274  qed
7275  AOT_hence ‹∃y(A!y & S = [λz [R]y] & ¬y[S])›
7276    by (rule a_prop[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1), rotated])
7277  then AOT_obtain b where b_prop: ‹A!b & S = [λz [R]b] & ¬b[S]›
7278    using "∃E"[rotated] by blast
7279  AOT_have 1: ‹a ≠ b›
7280    apply (rule "≡dfI"[OF "=-infix"])
7281    using a_enc b_prop[THEN "&E"(2)]
7282    using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7283  AOT_have a: ‹[λ [R]a] = ([R]a)›
7284    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7285    by (meson "log-prop-prop:2")
7286  AOT_have b: ‹[λ [R]b] = ([R]b)›
7287    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7288    by (meson "log-prop-prop:2")
7289  AOT_have ‹[λ [R]a] = [λ [R]b]›
7290    apply (rule "rule=E"[rotated, OF a[THEN id_sym]])
7291    apply (rule "rule=E"[rotated, OF b[THEN id_sym]])
7292    apply (rule "identity:4"[THEN "≡dfI", OF "&I", rotated])
7293     apply (rule "rule=E"[rotated, OF S_def])
7294    using b_prop "&E" apply blast
7295    apply (safe intro!: "&I")
7296    by (simp add: "log-prop-prop:2")+
7297  AOT_hence ‹A!a & A!b & a ≠ b & [λ [R]a] = [λ [R]b]›
7298    using 1 a_prop[THEN "&E"(1)] b_prop[THEN "&E"(1), THEN "&E"(1)] "&I" by auto
7299  AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λ [R]a] = [λ [R]y])› by (rule "∃I")
7300  AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λ [R]x] = [λ [R]y])› by (rule "∃I")
7301qed
7302
7303AOT_theorem aclassical2: ‹∃x∃y (A!x & A!y & x ≠ y & ∀F ([F]x ≡ [F]y))›
7304proof -
7305  AOT_have ‹∃x ∃y ([A!]x & [A!]y & x ≠ y &
7306               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7307    by (rule "aclassical:1"[THEN "∀E"(1)[where τ="«[λxy ∀F ([F]x ≡ [F]y)]»"]])
7308       "cqt:2[lambda]"
7309  then AOT_obtain x where ‹∃y ([A!]x & [A!]y & x ≠ y &
7310               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7311    using "∃E"[rotated] by blast
7312  then AOT_obtain y where 0: ‹([A!]x & [A!]y & x ≠ y &
7313               [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7314    using "∃E"[rotated] by blast
7315  AOT_have ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zx]x›
7316    apply (rule "β←C"(1))
7317      apply "cqt:2[lambda]"
7318     apply (fact "cqt:2[const_var]"[axiom_inst])
7319    apply (rule "β←C"(1))
7320      apply "cqt:2[lambda]"
7321    apply (simp add: "&I" "ex:1:a" prod_denotesI "rule-ui:3")
7322    by (simp add: "oth-class-taut:3:a" "universal-cor")
7323  AOT_hence ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zy]x›
7324    by (rule "rule=E"[rotated, OF 0[THEN "&E"(2)]])
7325  AOT_hence ‹[λxy ∀F ([F]x ≡ [F]y)]xy›
7326    by (rule "β→C"(1))
7327  AOT_hence ‹∀F ([F]x ≡ [F]y)›
7328    using "β→C"(1) old.prod.case by fast
7329  AOT_hence ‹[A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y)› using 0 "&E" "&I" by blast
7330  AOT_hence ‹∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I")
7331  AOT_thus ‹∃x∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I"(2))
7332qed
7333
7334AOT_theorem "kirchner-thm:1": ‹[λx φ{x}]↓ ≡ □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7335proof(rule "≡I"; rule "→I")
7336  AOT_assume ‹[λx φ{x}]↓›
7337  AOT_hence ‹□[λx φ{x}]↓› by (metis "exist-nec" "vdash-properties:10")
7338  moreover AOT_have ‹□[λx φ{x}]↓ → □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7339  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7340    AOT_modally_strict {
7341      fix x y
7342      AOT_assume 0: ‹[λx φ{x}]↓›
7343      moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7344      ultimately AOT_have ‹[λx φ{x}]x ≡ [λx φ{x}]y›
7345        using "∀E" by blast
7346      AOT_thus ‹(φ{x} ≡ φ{y})›
7347        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7348    }
7349  qed
7350  ultimately AOT_show ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7351    using "→E" by blast
7352next
7353  AOT_have ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y})) → □∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7354  proof(rule "RM:1"; rule "→I"; rule GEN)
7355    AOT_modally_strict {
7356      AOT_assume ‹∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7357      AOT_hence indisc: ‹φ{x} ≡ φ{y}› if ‹∀F([F]x ≡ [F]y)› for x y
7358        using "∀E"(2) "→E" that by blast
7359      AOT_show ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})› for y
7360      proof (rule "raa-cor:1")
7361        AOT_assume ‹¬(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7362        AOT_hence ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}) ∨ (¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7363          using "≡E"(1) "oth-class-taut:4:h" by blast
7364        moreover {
7365          AOT_assume 0: ‹∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}›
7366          AOT_obtain a where ‹∀F([F]a ≡ [F]y) & φ{a}›
7367            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7368          AOT_hence ‹φ{y}› using indisc[THEN "≡E"(1)] "&E" by blast
7369          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7370        }
7371        moreover {
7372          AOT_assume 0: ‹(¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7373          AOT_hence ‹∀x ¬(∀F([F]x ≡ [F]y) & φ{x})›
7374            using "&E"(1) "cqt-further:4" "→E" by blast
7375          AOT_hence ‹¬(∀F([F]y ≡ [F]y) & φ{y})› using "∀E" by blast
7376          AOT_hence ‹¬∀F([F]y ≡ [F]y) ∨ ¬φ{y}›
7377            using "≡E"(1) "oth-class-taut:5:c" by blast
7378          moreover AOT_have ‹∀F([F]y ≡ [F]y)› by (simp add: "oth-class-taut:3:a" "universal-cor")
7379          ultimately AOT_have ‹¬φ{y}› by (metis "¬¬I" "∨E"(2))
7380          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7381        }
7382        ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7383      qed
7384    }
7385  qed
7386  moreover AOT_assume ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7387  ultimately AOT_have ‹□∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7388    using "→E" by blast
7389  AOT_thus ‹[λx φ{x}]↓›
7390    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7391qed
7392
7393AOT_theorem "kirchner-thm:2": ‹[λx1...xn φ{x1...xn}]↓ ≡ □∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7394proof(rule "≡I"; rule "→I")
7395  AOT_assume ‹[λx1...xn φ{x1...xn}]↓›
7396  AOT_hence ‹□[λx1...xn φ{x1...xn}]↓› by (metis "exist-nec" "vdash-properties:10")
7397  moreover AOT_have ‹□[λx1...xn φ{x1...xn}]↓ → □∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7398  proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7399    AOT_modally_strict {
7400      fix x1xn y1yn :: ‹'a AOT_var›
7401      AOT_assume 0: ‹[λx1...xn φ{x1...xn}]↓›
7402      moreover AOT_assume ‹∀F([F]x1...xn ≡ [F]y1...yn)›
7403      ultimately AOT_have ‹[λx1...xn φ{x1...xn}]x1...xn ≡ [λx1...xn φ{x1...xn}]y1...yn›
7404        using "∀E" by blast
7405      AOT_thus ‹(φ{x1...xn} ≡ φ{y1...yn})›
7406        using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7407    }
7408  qed
7409  ultimately AOT_show ‹□∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7410    using "→E" by blast
7411next
7412  AOT_have ‹□(∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))) →
7413            □∀y1...∀yn((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7414  proof(rule "RM:1"; rule "→I"; rule GEN)
7415    AOT_modally_strict {
7416      AOT_assume ‹∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7417      AOT_hence indisc: ‹φ{x1...xn} ≡ φ{y1...yn}› if ‹∀F([F]x1...xn ≡ [F]y1...yn)› for x1xn y1yn
7418        using "∀E"(2) "→E" that by blast
7419      AOT_show ‹(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn}› for y1yn
7420      proof (rule "raa-cor:1")
7421        AOT_assume ‹¬((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7422        AOT_hence ‹((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}) ∨
7423                    (¬(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & φ{y1...yn})›
7424          using "≡E"(1) "oth-class-taut:4:h" by blast
7425        moreover {
7426          AOT_assume 0: ‹(∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) & ¬φ{y1...yn}›
7427          AOT_obtain a1an where ‹∀F([F]a1...an ≡ [F]y1...yn) & φ{a1...an}›
7428            using "∃E"[rotated, OF 0[THEN "&E"(1)]]  by blast
7429          AOT_hence ‹φ{y1...yn}› using indisc[THEN "≡E"(1)] "&E" by blast
7430          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7431        }
7432        moreover {
7433          AOT_assume 0: ‹(¬((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn}))) & φ{y1...yn})›
7434          AOT_hence ‹∀x1...∀xn ¬(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})›
7435            using "&E"(1) "cqt-further:4" "→E" by blast
7436          AOT_hence ‹¬(∀F([F]y1...yn ≡ [F]y1...yn) & φ{y1...yn})› using "∀E" by blast
7437          AOT_hence ‹¬∀F([F]y1...yn ≡ [F]y1...yn) ∨ ¬φ{y1...yn}›
7438            using "≡E"(1) "oth-class-taut:5:c" by blast
7439          moreover AOT_have ‹∀F([F]y1...yn ≡ [F]y1...yn)›
7440            by (simp add: "oth-class-taut:3:a" "universal-cor")
7441          ultimately AOT_have ‹¬φ{y1...yn}› by (metis "¬¬I" "∨E"(2))
7442          AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7443        }
7444        ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7445      qed
7446    }
7447  qed
7448  moreover AOT_assume ‹□∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7449  ultimately AOT_have ‹□∀y1...∀yn((∃x1...∃xn(∀F([F]x1...xn ≡ [F]y1...yn) & φ{x1...xn})) ≡ φ{y1...yn})›
7450    using "→E" by blast
7451  AOT_thus ‹[λx1...xn φ{x1...xn}]↓›
7452    by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7453qed
7454
7455AOT_theorem "kirchner-thm-cor:1": ‹[λx φ{x}]↓ → ∀x∀y(∀F([F]x ≡ [F]y) → □(φ{x} ≡ φ{y}))›
7456proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7457  fix x y
7458  AOT_assume ‹[λx φ{x}]↓›
7459  AOT_hence ‹□∀x∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7460    by (rule "kirchner-thm:1"[THEN "≡E"(1)])
7461  AOT_hence ‹∀x□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7462    using CBF[THEN "→E"] by blast
7463  AOT_hence ‹□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7464    using "∀E" by blast
7465  AOT_hence ‹∀y □(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7466    using CBF[THEN "→E"] by blast
7467  AOT_hence ‹□(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7468    using "∀E" by blast
7469  AOT_hence ‹□∀F ([F]x ≡ [F]y) → □(φ{x} ≡ φ{y})›
7470    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7471  moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7472  ultimately AOT_show ‹□(φ{x} ≡ φ{y})› using "→E" "ind-nec" by blast
7473qed
7474
7475AOT_theorem "kirchner-thm-cor:2":
7476  ‹[λx1...xn φ{x1...xn}]↓ → ∀x1...∀xn∀y1...∀yn(∀F([F]x1...xn ≡ [F]y1...yn) → □(φ{x1...xn} ≡ φ{y1...yn}))›
7477proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7478  fix x1xn y1yn
7479  AOT_assume ‹[λx1...xn φ{x1...xn}]↓›
7480  AOT_hence 0: ‹□∀x1...∀xn∀y1...∀yn (∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7481    by (rule "kirchner-thm:2"[THEN "≡E"(1)])
7482  AOT_have ‹∀x1...∀xn∀y1...∀yn □(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7483  proof(rule GEN; rule GEN)
7484    fix x1xn y1yn
7485    AOT_show ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7486      apply (rule "RM:1"[THEN "→E", rotated, OF 0]; rule "→I")
7487      using "∀E" by blast
7488  qed
7489  AOT_hence ‹∀y1...∀yn □(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7490    using "∀E" by blast
7491  AOT_hence ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7492    using "∀E" by blast
7493  AOT_hence ‹□(∀F ([F]x1...xn ≡ [F]y1...yn) → (φ{x1...xn} ≡ φ{y1...yn}))›
7494    using "∀E" by blast
7495  AOT_hence 0: ‹□∀F ([F]x1...xn ≡ [F]y1...yn) → □(φ{x1...xn} ≡ φ{y1...yn})›
7496    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7497  moreover AOT_assume ‹∀F([F]x1...xn ≡ [F]y1...yn)›
7498  moreover AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]↓› by "cqt:2[lambda]"
7499  ultimately AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]x1...xn ≡ [λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]y1...yn›
7500    using "∀E" by blast
7501  moreover AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]y1...yn›
7502    apply (rule "β←C"(1))
7503      apply "cqt:2[lambda]"
7504     apply (fact "cqt:2[const_var]"[axiom_inst])
7505    by (simp add: RN GEN "oth-class-taut:3:a")
7506  ultimately AOT_have ‹[λx1...xn □∀F ([F]x1...xn ≡ [F]y1...yn)]x1...xn› using "≡E"(2) by blast
7507  AOT_hence ‹□∀F ([F]x1...xn ≡ [F]y1...yn)›
7508    using "β→C"(1) by blast
7509  AOT_thus ‹□(φ{x1...xn} ≡ φ{y1...yn})› using "→E" 0 by blast
7510qed
7511
7512AOT_define propositional :: ‹Π ⇒ φ› (‹Propositional'(_')›)
7513  "prop-prop1": ‹Propositional([F]) ≡df ∃p(F = [λy p])›
7514
7515AOT_theorem "prop-prop2:1": ‹∀p [λy p]↓›
7516  by (rule GEN) "cqt:2[lambda]"
7517
7518AOT_theorem "prop-prop2:2": ‹[λν φ]↓›
7519  by "cqt:2[lambda]"
7520
7521AOT_theorem "prop-prop2:3": ‹F = [λy p] → □∀x([F]x ≡ p)›
7522proof (rule "→I")
7523  AOT_assume 0: ‹F = [λy p]›
7524  AOT_show ‹□∀x([F]x ≡ p)›
7525    by (rule "rule=E"[rotated, OF 0[symmetric]]; rule RN; rule GEN; rule "beta-C-meta"[THEN "→E"])
7526      "cqt:2[lambda]"
7527qed
7528
7529AOT_theorem "prop-prop2:4": ‹Propositional([F]) → □Propositional([F])›
7530proof(rule "→I")
7531  AOT_assume ‹Propositional([F])›
7532  AOT_hence ‹∃p(F = [λy p])› using "≡dfE"[OF "prop-prop1"] by blast
7533  then AOT_obtain p where ‹F = [λy p]› using "∃E"[rotated] by blast
7534  AOT_hence ‹□(F = [λy p])› using "id-nec:2" "modus-tollens:1" "raa-cor:3" by blast
7535  AOT_hence ‹∃p □(F = [λy p])› using "∃I" by fast
7536  AOT_hence 0: ‹□∃p (F = [λy p])› by (metis Buridan "vdash-properties:10")
7537  AOT_thus ‹□Propositional([F])›
7538    using "prop-prop1"[THEN "≡Df"]
7539    by (AOT_subst ‹Propositional([F])› ‹∃p (F = [λy p])›) auto
7540qed
7541
7542AOT_define indicriminate :: ‹Π ⇒ φ› ("Indiscriminate'(_')")
7543  "prop-indis": ‹Indiscriminate([F]) ≡df F↓ & □(∃x [F]x → ∀x [F]x)›
7544
7545AOT_theorem "prop-in-thm": ‹Propositional([Π]) → Indiscriminate([Π])›
7546proof(rule "→I")
7547  AOT_assume ‹Propositional([Π])›
7548  AOT_hence ‹∃p Π = [λy p]› using "≡dfE"[OF "prop-prop1"] by blast
7549  then AOT_obtain p where Π_def: ‹Π = [λy p]› using "∃E"[rotated] by blast
7550  AOT_show ‹Indiscriminate([Π])›
7551  proof (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7552    AOT_show ‹Π↓›
7553      using Π_def by (meson "t=t-proper:1" "vdash-properties:6")
7554  next
7555    AOT_show ‹□(∃x [Π]x → ∀x [Π]x)›
7556    proof (rule "rule=E"[rotated, OF Π_def[symmetric]]; rule RN; rule "→I"; rule GEN)
7557      AOT_modally_strict {
7558        AOT_assume ‹∃x [λy p]x›
7559        then AOT_obtain a where ‹[λy p]a› using "∃E"[rotated] by blast
7560        AOT_hence 0: ‹p› by (metis "β→C"(1))
7561        AOT_show ‹[λy p]x› for x
7562          apply (rule "β←C"(1))
7563            apply "cqt:2[lambda]"
7564           apply (fact "cqt:2[const_var]"[axiom_inst])
7565          by (fact 0)
7566      }
7567    qed
7568  qed
7569qed
7570
7571AOT_theorem "prop-in-f:1": ‹Necessary([F]) → Indiscriminate([F])›
7572proof (rule "→I")
7573  AOT_assume ‹Necessary([F])›
7574  AOT_hence 0: ‹□∀x1...∀xn [F]x1...xn› using "≡dfE"[OF "contingent-properties:1"] by blast
7575  AOT_show ‹Indiscriminate([F])›
7576    by (rule "≡dfI"[OF "prop-indis"])
7577       (metis "0" "KBasic:1" "&I" "ex:1:a" "rule-ui:2[const_var]" "vdash-properties:6") 
7578qed
7579
7580AOT_theorem "prop-in-f:2": ‹Impossible([F]) → Indiscriminate([F])›
7581proof (rule "→I")
7582  AOT_modally_strict {
7583    AOT_have ‹∀x ¬[F]x → (∃x [F]x → ∀x [F]x)›
7584      by (metis "instantiation" "cqt-orig:3" "Hypothetical Syllogism" "deduction-theorem" "raa-cor:3")
7585  }
7586  AOT_hence 0: ‹□∀x ¬[F]x → □(∃x [F]x → ∀x [F]x)›
7587    by (rule "RM:1")
7588  AOT_assume ‹Impossible([F])›
7589  AOT_hence ‹□∀x ¬[F]x› using "≡dfE"[OF "contingent-properties:2"] "&E" by blast
7590  AOT_hence 1: ‹□(∃x [F]x → ∀x [F]x)› using 0 "→E" by blast
7591  AOT_show ‹Indiscriminate([F])›
7592    by (rule "≡dfI"[OF "prop-indis"]; rule "&I")
7593       (simp add: "ex:1:a" "rule-ui:2[const_var]" 1)+
7594qed
7595
7596AOT_theorem "prop-in-f:3:a": ‹¬Indiscriminate([E!])›
7597proof(rule "raa-cor:2")
7598  AOT_assume ‹Indiscriminate([E!])›
7599  AOT_hence 0: ‹□(∃x [E!]x → ∀x [E!]x)›
7600    using "≡dfE"[OF "prop-indis"] "&E" by blast
7601  AOT_hence ‹◇∃x [E!]x → ◇∀x [E!]x›
7602    using "KBasic:13" "vdash-properties:10" by blast
7603  moreover AOT_have ‹◇∃x [E!]x›
7604    by (simp add: "thm-cont-e:3")
7605  ultimately AOT_have ‹◇∀x [E!]x›
7606    by (metis "vdash-properties:6")
7607  AOT_thus ‹p & ¬p› for p
7608    by (metis "≡dfE" "conventions:5" "o-objects-exist:5" "reductio-aa:1")
7609qed
7610
7611AOT_theorem "prop-in-f:3:b": ‹¬Indiscriminate([E!]-)›
7612proof (rule "rule=E"[rotated, OF "rel-neg-T:2"[symmetric]]; rule "raa-cor:2")
7613  AOT_assume ‹Indiscriminate([λx ¬[E!]x])›
7614  AOT_hence 0: ‹□(∃x [λx ¬[E!]x]x → ∀x [λx ¬[E!]x]x)›
7615    using "≡dfE"[OF "prop-indis"] "&E" by blast
7616  AOT_hence ‹□∃x [λx ¬[E!]x]x → □∀x [λx ¬[E!]x]x›
7617    using "→E" "qml:1" "vdash-properties:1[2]" by blast
7618  moreover AOT_have ‹□∃x [λx ¬[E!]x]x›
7619    apply (AOT_subst ‹[λx ¬E!x]x› ‹¬E!x› bound: x)
7620    apply (rule "beta-C-meta"[THEN "→E"])
7621     apply "cqt:2[lambda]"
7622    by (metis (full_types) "B◇" RN "T◇" "cqt-further:2" "o-objects-exist:5" "vdash-properties:10")
7623  ultimately AOT_have 1: ‹□∀x [λx ¬[E!]x]x›
7624    by (metis "vdash-properties:6")
7625  AOT_hence ‹□∀x ¬[E!]x›
7626    by (AOT_subst (reverse) ‹¬[E!]x›  ‹[λx ¬[E!]x]x› bound: x)
7627       (auto intro!: "cqt:2" "beta-C-meta"[THEN "→E"])
7628  AOT_hence ‹∀x □¬[E!]x› by (metis "CBF" "vdash-properties:10")
7629  moreover AOT_obtain a where abs_a: ‹O!a›
7630    using "instantiation" "o-objects-exist:1" "qml:2" "vdash-properties:1[2]" "vdash-properties:6" by blast
7631  ultimately AOT_have ‹□¬[E!]a› using "∀E" by blast
7632  AOT_hence 2: ‹¬◇[E!]a› by (metis "≡dfE" "conventions:5" "reductio-aa:1")
7633  AOT_have ‹A!a›
7634    apply (rule "=dfI"(2)[OF AOT_abstract])
7635     apply "cqt:2[lambda]"
7636    apply (rule "β←C"(1))
7637      apply "cqt:2[lambda]"
7638    using "cqt:2[const_var]"[axiom_inst] apply blast
7639    by (fact 2)
7640  AOT_thus ‹p & ¬p› for p using abs_a
7641    by (metis "≡E"(1) "oa-contingent:2" "reductio-aa:1")
7642qed
7643
7644AOT_theorem "prop-in-f:3:c": ‹¬Indiscriminate(O!)›
7645proof(rule "raa-cor:2")
7646  AOT_assume ‹Indiscriminate(O!)›
7647  AOT_hence 0: ‹□(∃x O!x → ∀x O!x)›
7648    using "≡dfE"[OF "prop-indis"] "&E" by blast
7649  AOT_hence ‹□∃x O!x → □∀x O!x›
7650    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7651  moreover AOT_have ‹□∃x O!x›
7652    using "o-objects-exist:1" by blast
7653  ultimately AOT_have ‹□∀x O!x›
7654    by (metis "vdash-properties:6")
7655  AOT_thus ‹p & ¬p› for p
7656    by (metis "o-objects-exist:3" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7657qed
7658
7659AOT_theorem "prop-in-f:3:d": ‹¬Indiscriminate(A!)›
7660proof(rule "raa-cor:2")
7661  AOT_assume ‹Indiscriminate(A!)›
7662  AOT_hence 0: ‹□(∃x A!x → ∀x A!x)›
7663    using "≡dfE"[OF "prop-indis"] "&E" by blast
7664  AOT_hence ‹□∃x A!x → □∀x A!x›
7665    using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7666  moreover AOT_have ‹□∃x A!x›
7667    using "o-objects-exist:2" by blast
7668  ultimately AOT_have ‹□∀x A!x›
7669    by (metis "vdash-properties:6")
7670  AOT_thus ‹p & ¬p› for p
7671    by (metis "o-objects-exist:4" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7672qed
7673
7674AOT_theorem "prop-in-f:4:a": ‹¬Propositional(E!)›
7675  using "modus-tollens:1" "prop-in-f:3:a" "prop-in-thm" by blast
7676
7677AOT_theorem "prop-in-f:4:b": ‹¬Propositional(E!-)›
7678  using "modus-tollens:1" "prop-in-f:3:b" "prop-in-thm" by blast
7679
7680AOT_theorem "prop-in-f:4:c": ‹¬Propositional(O!)›
7681  using "modus-tollens:1" "prop-in-f:3:c" "prop-in-thm" by blast
7682
7683AOT_theorem "prop-in-f:4:d": ‹¬Propositional(A!)›
7684  using "modus-tollens:1" "prop-in-f:3:d" "prop-in-thm" by blast
7685
7686AOT_theorem "prop-prop-nec:1": ‹◇∃p (F = [λy p]) → ∃p(F = [λy p])›
7687proof(rule "→I")
7688  AOT_assume ‹◇∃p (F = [λy p])›
7689  AOT_hence ‹∃p ◇(F = [λy p])›
7690    by (metis "BF◇" "vdash-properties:10")
7691  then AOT_obtain p where ‹◇(F = [λy p])› using "∃E"[rotated] by blast
7692  AOT_hence ‹F = [λy p]› by (metis "derived-S5-rules:2" emptyE "id-nec:2" "vdash-properties:6")
7693  AOT_thus ‹∃p(F = [λy p])› by (rule "∃I")
7694qed
7695
7696AOT_theorem "prop-prop-nec:2": ‹∀p (F ≠ [λy p]) → □∀p(F ≠ [λy p])›
7697proof(rule "→I")
7698  AOT_assume ‹∀p (F ≠ [λy p])›
7699  AOT_hence ‹(F ≠ [λy p])› for p
7700    using "∀E" by blast
7701  AOT_hence ‹□(F ≠ [λy p])› for p
7702    by (rule "id-nec2:2"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7703  AOT_hence ‹∀p □(F ≠ [λy p])› by (rule GEN)
7704  AOT_thus ‹□∀p (F ≠ [λy p])› using BF[THEN "→E"] by fast
7705qed
7706
7707AOT_theorem "prop-prop-nec:3": ‹∃p (F = [λy p]) → □∃p(F = [λy p])›
7708proof(rule "→I")
7709  AOT_assume ‹∃p (F = [λy p])›
7710  then AOT_obtain p where ‹(F = [λy p])› using "∃E"[rotated] by blast
7711  AOT_hence ‹□(F = [λy p])› by (metis "id-nec:2" "vdash-properties:6")
7712  AOT_hence ‹∃p□(F = [λy p])› by (rule "∃I")
7713  AOT_thus ‹□∃p(F = [λy p])› by (metis Buridan "vdash-properties:10")
7714qed
7715
7716AOT_theorem "prop-prop-nec:4": ‹◇∀p (F ≠ [λy p]) → ∀p(F ≠ [λy p])›
7717proof(rule "→I")
7718  AOT_assume ‹◇∀p (F ≠ [λy p])›
7719  AOT_hence ‹∀p ◇(F ≠ [λy p])› by (metis "Buridan◇" "vdash-properties:10")
7720  AOT_hence ‹◇(F ≠ [λy p])› for p
7721    using "∀E" by blast
7722  AOT_hence ‹F ≠ [λy p]› for p
7723    by (rule "id-nec2:3"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7724  AOT_thus ‹∀p (F ≠ [λy p])› by (rule GEN)
7725qed
7726
7727AOT_theorem "enc-prop-nec:1": ‹◇∀F (x[F] → ∃p(F = [λy p])) → ∀F(x[F] → ∃p (F = [λy p]))›
7728proof(rule "→I"; rule GEN; rule "→I")
7729  fix F
7730  AOT_assume ‹◇∀F (x[F] → ∃p(F = [λy p]))›
7731  AOT_hence ‹∀F ◇(x[F] → ∃p(F = [λy p]))›
7732    using "Buridan◇" "vdash-properties:10" by blast
7733  AOT_hence 0: ‹◇(x[F] → ∃p(F = [λy p]))› using "∀E" by blast
7734  AOT_assume ‹x[F]›
7735  AOT_hence ‹□x[F]› by (metis "en-eq:2[1]" "≡E"(1))
7736  AOT_hence ‹◇∃p(F = [λy p])›
7737    using 0 by (metis "KBasic2:4" "≡E"(1) "vdash-properties:10")
7738  AOT_thus ‹∃p(F = [λy p])›
7739    using "prop-prop-nec:1"[THEN "→E"] by blast
7740qed
7741
7742AOT_theorem "enc-prop-nec:2": ‹∀F (x[F] → ∃p(F = [λy p])) → □∀F(x[F] → ∃p (F = [λy p]))›
7743  using "derived-S5-rules:1"[where Γ="{}", simplified, OF "enc-prop-nec:1"]
7744  by blast
7745
7746(*<*)
7747end
7748(*>*)